An idea that comes into mind is to add an attachment to your player as to where the pet should start it’s animation (AKA, just the position of where the pet is on your mounting animation relative to the player).
And then whenever the pet gets close enough, add a WeldConstraint on the player with Part0 = partOfPlayer and Part1 = partOfPet to lock it in place and play the mounting animation.
Here’s a video showcasing what I mean:
testfile.rbxm (20.4 KB) – you can mess around with this to see how it works.