The turn player can activate Quick-Play Spell Cards from their hand during any Phase of their turn; either player can activate Set Quick-Play Spell cards during any Phase in either player's turn, except during the turn they are Set.

I set the Twin Twisters on my turn. Wait until opponent sets cards. I try to activate during his End Phase, or at least Main Phase 2, but nope 😛

It might be a mistake of mine, but judging from the rules it isn't(?)
Hold left-click or "A" to force YGOPRO to ask for activation of cards, because it sometimes doesn't.
The Mother Toast wrote:

This be the day of days! Rejoice my toasty brethren!