Oh, I understand the distances well enough. I live in Australia, not the UK. We have to suffer long distances to do anything here. Much worse than the US. My dad cycled around Australia in his younger years as well as a full lap of Europe and most of South America. I totally understand the convenience of cars over bicycles and while it may seem a crazy intrepid idea to Americans, really it is not that big a deal if you have a couple of months to travel around, want a real adventure and are physically capable. Not to mention it can save a ton of money!
National park "hopping"
I agree but the perspective I'm coming from is that of hopping from National Park to National Park bicycling Across America is not that uncommon and a fair number of people do it but hopping from national park to National Park is a whole different kind of vacation
I think budget is a big part of what you are trying to do, and I do think you can save some money by camping, but by in large the other comments on this thread are right It's not easy to get from park to park cheaply unless you plan either on taking a large amount of time in transit (which sometimes can be fun) or don't mind spending some extra money to transit more quickly.
But I'll skip over that for now and suggest maybe based upon your summer timing you focus on the West Coast, or the mid-Atlantic-Northeast section of the US. The biggest advantage of doing the east coast trail, would be being able to hike large portions of the time along the Appalachian trail, which won't leave you huge distances from many state parks and a few of the National Parks which you'd have to transit to, but it would allow you to hike and camp a lot more and the trail does pass though several parks. You'd end up having to settle for one of the "Grand Canyon's of the east" (as several parks bill themselves), and probably wouldn't see as much in the way of larger animals (you'd see deer and maybe bear, and maybe even some moose if you got up into Maine)
It would mean going to a number of lesser known, and smaller parks, and doing a bit more research than just doing the more well known parks but that might just make it a bit more special.
There also are a number of cheap bus lines, some better than others, that run from the Carolina's up to Boston from the major cities (look around in almost any city that has a good sized Chinatown) plus companies like Megabus and Bolt Bus (the fares on those two are cheapest the farther in advance you book them). They won't be the most comfortable but they do work. You'd still need to get to the parks themselves - but some you can get to by hiking, biking, others you can't.
How doable it is would really depend on your tolerance for travel time and how much you'd be willing to trade in exchange for time in said parks.
Hope that helps a little.