I think the parenthetical is necessary. Here is my interpretation.
Breaking it down, Prince says:
1. Set aside an Action card from your hand costing up to $4.
2. At the start of each of your turns, play that Action.
3. Set that action aside again when you discard it from play.
4. Stop playing that action if you fail to set it aside on a turn you play it.
1 determines the princed action.
2 tells you to play that action repeatedly.
3 tells you when to set it aside again, for the purpose of bookkeeping.
4 tells you when the action stops getting played.
Let's consider what happens with Caravan.
On the turn you play Prince, you set aside Caravan according to 1.
On the second turn, you play Caravan according to 2.
At the end of that turn, you do not discard Caravan so 3 doesn't even trigger. Caravan does not get set aside.
Because you fail to set it aside on this turn on which you played Caravan, 4 stops Prince from playing Caravan again.
If you remove the parenthetical, then there is no 4 and Prince never says to stop playing Caravan. Number 3 does not say "stop". Even though you fail to set it aside, Caravan will continue to be played. Instead of the above, the following would happen:
1. Set Caravan aside with Prince on turn 1.
2. Play Caravan on turn 2.
3. Get duration effects of Caravan AND play it again. Repeat this step for every subsequent turn.
It's only because of the parenthetical that you need to set aside the card to play it again with Prince. If not for that, the card would get played no matter what. Even if the card is already in play or if the card trashed itself, it would still get played again, just like it works with Throne Room when the throned card is already in play (most cases) or when it trashes itself.