It's easily do-able... but two parks is far more practical. You can do it if one of your park drops is just doing 2 or 3 things.. but you might find yourself a bit ragged by the end of the day. Disney's new FP+ system will be working against you as well.
It wasn't uncommon for us to do something like.. hit DHS first thing in the morning to get our 'must fixes' in for our favorites.. then hit another park for the day... and then hit MK after dinner time to see fireworks and do late night rides. We would do this with AK as well.. go in and ride everest, then leave, etc.
I would only suggest this tho if you have people that have already seen the parks and know what they want to do or not... if you want to explore/experience the parks... this is not the model to use. This is for people who are seeking to do certain attractions and not much else.
Exactly right. If the people in your party have experienced the parks before and you are just trying to hit a few attractions in each, this is do-able, but you won't really get to experience any of the parks in any detail and you will most certainly exhaust yourselves. My family are runners so we tend to do parks at a high energy level/rate of speed, but I doubt we would try a day this ambitious - we like to go-go-go so we can then sit and relax.
But, of course, seeing your idea, I had to think of my plan if I were trying to do something like that. I'd book FP+ for Epcot - Soarin' (or Test Track), Character Spot, and Mission:Space. I'd get to Studios at opening and go straight to Toy Story, then to RnRC and Tower of Terror. Picking up some baked goods at Starring Rolls and ideally out of the park by 11 or 11:30am.
Into Epcot by noon, I'd hope to do Soarin' with FP+ right away, then Living with the Land (always a must-do for me), Character Spot with FP+ (gotta fit some character time into the day for my daughter!), back to World Showcase for a late counter lunch in UK, France, or Germany with a couple of drinks - eating at non-usual meal times means shorter lines and faster service! Then back to Future World for FP+ at Mission:Space and maybe Test Track single rider (although last few times I've seen it, the line for single rider has not been very beneficial). Out of Epcot by 5pm or so.
To MK - odds are, not actually in the park until after 6pm. As soon as I get in the park, I'm hitting a FP+ kiosk to see if I can get anything booked. Whatever I can get booked, I'm going to do (trying to keep things in the same area) and then staying in the park for the entire evening. Sometimes later in the evening some of the lines for the more family friendly rides can go down a little bit and I'd hope to take advantage of that.
Sadly, late June means that afternoon thunderstorms are a definite possibility - plan for them with ponchos, because you can't hide out if you are trying to hit three parks in one day! Luckily, if it is a real soaker, Magic Kingdom will clear out quite a bit and you might be able to hit some more attractions!
Good luck and maybe we'll see you there (visiting June 28-July 7).