Having done Yellowstone many times over the past 30 or so years, I would say to avoid it during April & even up through mid May. We spent part of our honeymoon in Yellowstone at the end of April/beginning of May and there was still 4+ feet of snow in most areas. While there, we encountered a snowstorm that closed several roads in the park. Even in mid June we've encountered snow in some areas. With the exception of Mammoth, which is open year round, the campgrounds start opening in early May - usually Madison is the first, with others following in mid to late May. We've also been there in August...mid August is really busy but late August, not so much as the tourist season is winding down.
We stopped at Crater Lake in mid July last summer...temps were in the 40's with a 30mph wind...still a lot of snow combined with mosquitoes! We also stopped at Lassen Volcanic NP. Weather was nice but cool...still quite a bit of snow in several areas.
We've spent quite a bit of time in the national parks in Utah and have found mid June a good time, weather-wise, for camping...not too hot, not too cold. Same with NPs in Arizona & New Mexico. Camped in the Grand Canyon at the end of April...was pretty chilly with 20º at night plus wind...we were tent campers then & I was 6 months pregnant, but we had a great time.
As far as places to stay when traveling point-to-point & trying to make miles, we have used Wal-Mart parking lots from Wisconsin to Florida, Minnesota to Idaho, Nevada to Nebraska & down to Texas. Never had any problems...always let them know we were there. Sometimes they would even show us a good spot. Many also would have security make rounds every so often throughout the night.
Sounds like a wonderful trip! So many beautiful parks to visit. Craig & I started dating after spending a day together at Yellowstone...that's why we decided it was only fitting to spend our honeymoon there. We made it a goal to visit/camp at as many national parks as we could in our lifetime together and so far we've checked off more than 1/2 of them over the past 30 years, plus many national monuments, landmarks & historic sites. Always well worth the visit.
Have a great trip!