I have to ask... what *if* you modified your original question as follows:
Paladin with Ride-By-Attack mounted on Celestial Tiger which has the feat Spring Attack...
How many attacks can they do? Probably only two, or one each, but the mount can move it's full movement, and the two attacks can occur anywhere along the move, right? Now, is pounce listed as a standard action, move action, or as an attack? If it's just an attack, sure, use it (per the rules) but I suspect a close reading of 'pounce' will indicate it's not just an attack but more analogous to grapple, in which case I'd have a problem with seeing it used in conjunction with spring attack... I mean, really, move - grapple - move away???