Ok, so if a PC takes Ride-By Attack then only the PC can attack during the charge. The mount can't. But what if the MOUNT takes the feat? Is that allowed? And if so, would the rider by able to attack during the charge or would they both need to take it?
"When you are mounted and use the charge action," seems to suggest that your horse would need to be riding another horse while you ride it to do that. I was going to suggest maybe Spring Attack on the Horse and some minor GM fiat magic. But that'd require your GM allowing it to work (as spring attack doesn't work on a charge).
How about if I give the mount Fly-By Attack? It doesn't need a charge. And the mount is able to fly, of course.
How about if I give the mount Fly-By Attack? It doesn't say the person taking the feat be mounted.
It specifically states it must make a move action. When you charge, the mount also charges, thus that wouldn't work either.
Not with a charge though, just a regular move action.
You also normally cannot attack in the middle of a move. However, mounted combat rules fall short on clarifying this. However, the fact that ride-by attack exists suggests you can't normally attack in the middle of your mounts move (in fact, it even states for ranged that it has to be in the middle of the move; for double move and run).
So... What does that mean for you? Well, besides the fact that mounted combat rules need a rewrite? That you likely cannot have both you and the horse spring attack. I as a GM though, would probably just let you have your horse attack once. You are, after all, limiting yourself to one attack anyways and it's not like it's broken. But you'd need to speak to your GM to allow that.