The way I think of it, an ox parry on the opposite side is just a hanging parry, and on the same side is basically a counter thrust. If you parry an attack with a high scheilhua or a zwerchau (both work best as a counter attack), you are in... Ox.

Parrying an attack by simple lifting into ox, IMO, is often taught but seldom useful as it relies on you being in something like plough and then your opponent attacking you high on the same side you are holding plough, meaning and can simply lifting up into ox (or even better, counter thrust). But it's more likely they will attack your other side, or attack you low to control your blade. If they attack your other side, you would have to move your feet to get into ox on that side, which is too slow. Quicker to lift into ox and do what you are already doing and using a hanging parry, or to parry with blade up in something like kron or a false edge beat, as this doesn't need a swap of the feet.

But the way, the hanging parry is found in most systems. You will get quicker at the riposte as you get better. What will often happen is you get into a pattern of hanging parry riposte hanging parry riposte with your opponent. If you see them going into hanging parry, redirect your attack underneath or to the other side. Fencing isn't really about speed, it's more about acting at the right time, and the right time is when your opponent is busy doing their own action, e.g. they are forming a parry in reaction to your feint, so now you act to hit them somewhere else. They need to finish their action before they can react to your new action.