The best light for photography is fairly early in the morning before the sun gets too high, or in the last hour or two before sunset - what's known as "the golden hour" because of the beautiful golden tone to the sunlight. Think of how you squint in bright sunlight at noon, and the shadows that can come down your face (under your eyes, nose, lips, chin) when the sun is high overhead. Yes, a professional photographer worth the name will have the right equipment and know-how to work around the less-than-ideal lighting of midday, but for the best results (and you're paying enough to want the best, right?), I'd definitely suggest going with a late afternoon/pre-sunset photoshoot. Unless your family (and your photographer!) are cheerfully early risers and want to bang out the photos first thing... (When I pay for a professional photo session, I always book late afternoon.)
As for how to make that work, if you don't want a resort/mini golf/etc day, I'd say the best thing is to pick a park that you're going to more than once... or that you consider to be a half-day park... one you don't mind being at for a shorter time that day. Do your morning/lunch/early afternoon as you wish, leaving the park with enough time to freshen up (plus nap and probably grab a light snack, if you have littles who you want to keep from getting cranky) before you meet your photographer. You didn't say when you were going - obviously you'll need a bit more "freshen up" time in August than in, say, December, when you aren't as likely to be covered in sweat from a day at the parks!
I'm going to check out the recommended photographers as well, I think. A few years ago, we did a family photo session with the Disney photographers (once, at the Beach Club) and it came out great, but I have since heard that that's luck of the draw - that some photographers are better than others. That said, we did love the photographer/assistant who shot our pics back in 2011, so I'd consider requesting them again... if I could remember their names! (I'm the worst with names.)