Hmm, that’s unfortunate. I actually have a doctors note that I carry along which recommends the use of an assisted bicycle. I ride thoughtfully and have never been asked to show it. I cannot comment on the federal parks, perhaps they are not aware of the federal law classifying electric assist bikes that perform at or below 20 mph and 750 watts as bicycles. Having a note and this information would be a good response if you were questioned.
The release of Garmin’s new Edge 520 Plus GPS bike computer made a big splash in the world of cycling. It was an impressive upgrade, likely a response to increasing competition from the likes of Wahoo. But the Garmin Edge 520 Plus upgrade has caused a little confusion, because it seems to offer the same mapping and navigation capabilities as the more expensive Edge 820 and 1030. We clear up the confusion with two in-depth posts, Garmin Edge 520 vs 520 Plus, and Garmin Edge 1030 vs 820 vs 520 Plus.
There are individuals who claim to have lost considerable amounts of weight by using an electric bike. A recent prospective cohort study however found that people using e-bikes have a higher BMI. By making the biking terrain less of an issue, people who wouldn't otherwise consider biking can use the electric assistance when needed and otherwise pedal as they are able. This means people of lower fitness levels or who haven't cycled in many years can start enjoying the many health benefits E-bikes have to offer. 
Great question… a real world situation from someone doing their best to follow the law but also be realistic about time/distance. What follows is my opinion on the matter based on lots of electric bike riding experience, this is not legal advice. You can probably ride with electric assist across the dirt way with no issues and avoid any negative response that might arise as long as you’re polite to other cyclists and pedestrians and pedal along without going too fast. If someone called you out about it you could easily hop off, walk your bike and explain that you use the assist to help make your commute possible or possibly, like me, you have a knee injury and it’s difficult to pedal through softer terrain like gravel. Another different approach might be to shut the bike off completely before entering the gravel section, you could even take the battery off and put it in a backpack or as mentioned earlier… just walk the bike. Imagine if you had a motorized dirt bike and were just pushing it along a sidewalk… this kind of vehicle definitely is not allowed on sidewalks or most gravel paths like the one you’re describing but if you were escorting it carefully, you’d be honoring the spirit of the law and if an officer or pedestrian jumped out and started questioning you about your “motorized vehicle” it is my feeling that a genuine explanation and apology or request for guidance would go very well. I personally have never had issues riding electric bikes in part because I am thoughtful about how I use them. I do occasionally switch them off and sometimes I get off and walk. I have asked police in many cities across the US what they thought about ebikes and in every case I have received positive interest and support with guidance to ride safe with a helmet and follow traffic laws in the street. I hope this helps and I wish you well, it’s nice that you care enough to ask and I hope you’re treated well by others out on the road. The flip side of this response is that I have been harassed, yelled at and even swerved at by automobiles when riding bicycles and electric bikes. This usually happens in the evening after work lets out when traffic is heavy and I’m riding on the shoulder or in the street (where marked to do so) and I believe it has to do with territory, testosterone and socio-economic standing more so than laws or anything like that.
"Electric-assisted bicycle" means a bicycle with two or three wheels, a saddle, fully operative pedals for human propulsion, and an electric motor. The electric-assisted bicycle's electric motor must have a power output of no more than one thousand watts, be incapable of propelling the device at a speed of more than twenty miles per hour on level ground, and be incapable of further increasing the speed of the device when human power alone is used to propel the device beyond twenty miles per hour.
The Registrar will permit an electric motor driven cycle to be registered if it meets Canada Motor Vehicle Safety Standards (CMVSS) as a Limited Speed Motorcycle, or Scooter as is done with gas powered motor driven cycles. If the vehicle was manufactured after 1988 it will bear a compliance label stating that it meets these standards. The operator will be subject to all the requirements placed on operators of motor driven cycles.
The battery is the pedelec’s power source. It supplies the motor with the electrical energy that is required to provide power assistance when cycling. So it is hardly surprising that there is frequent discussion and “talking shop” about the eBike’s battery in particular. What is the difference between batteries? How far can you go on a fully charged battery? What do you have to remember about storage? Thomas Raica, head of technical customer application, here provides information and advice.
“Changes in production and consumption patterns are a crucial element of the sustainability agenda. Communication between product developers and users, and user integration in product development, can serve as a means for organizational as well as individual learning processes, resulting in sustainable product development. [...] Improved methods, such as INNOCOPE (innovating through consumer-integrated product development), tested in this study with a cycle manufacturer and resulting in a new product, a pedelec, are needed for effective communication, activating consumers and enabling them to promote sustainability goals. Through co-operative product development processes key factors facilitating and obstructing the adoption of sustainable innovations may be identified. Such processes can enhance the emergence and diffusion of sustainable product innovations and different forms and bodies of knowledge can be combined. Integrating users' contextual everyday knowledge of the product with the technical knowledge of companies may lead to mutual learning, technical innovations and changes in consumer behaviour.” (Hoffmann, 2007).
Electric motor assisted bicycles have been banned in the State of New York and are not permitted for on-road use. It appears the only known allowance of an electric bicycle is if it is an electric powered moped, at this time. There is a proposed bill to allow ebikes. As of May 2009, Bill A2393("Defines the term electric assisted bicycle") has been passed in the NY State Assembly and its corresponding Bill S4014, sponsored by Senator Thomas Morahan, is before the NY State Senate.
Built by a company that’s made cycling equipment for more than four decades, the Vado feels more like a traditional bicycle than almost any other ebike. Its frame and components have been tuned to provide a familiar experience, making it easy for new and long-time cyclists to jump on and start pedaling. Specialized’s heritage shines through nicely, helping separate itself from the competition in an increasingly crowded ebike market.
The other shout-out goes to Paul Bogaert who decided to close the Bike Doctor after 27 years and instead cycle tour the world with his wife. Paul had an early role in Vancouver bicycle advocacy and at bringing cycle riding to a less elitist/athletic and more everyday commuter crowd through their shop. They were among the early shops to embrace family friendly cargo bikes and appreciated the role that electric would play in making cycling more broadly appealing.
And let’s not forget the economic advantages of owning an e-bike. The annual cost of running a new family car is, on average, about $9,000 per year. Running an electric bike costs around $400 per year. And while filling a gas tank costs around $30, recharging an electric bike battery costs only about 50 cents. A tank of gas may get you further, but not 60 times further!