$33 Party, QQ post flop.

[RobGW]

So what are you planning to do when one of them bets out on the turn? Do you just assume he missed his check raise with an A and fold? By checking behind you are inducing a bet from a hand like KT or 88. You'll be folding the best hand too often then. Or do you call down? If he does have the A and missed his check raise you are now donating money to him. By betting this flop, you'll have more info to play the hand out.
Now for the river, he's shown no strength. He has yet to make a bet. He just called PF and on the turn. Since you checked behind on the flop, he's probably thinking you don't have the A therefore his T or PP is good. Especially with 2 A's out there. If he did have an A he probably would have bet the turn. Get the value bet in on the river.