2 points:
-You are right to want to get things in writing. Make sure you do that ASAP.
-Your parents are unlikely, IMO, to recover any lost rent from this time, so any agreement to repay extra money down the road is *probably* worthless in reality. That doesn't make it pointless to make the agreement, but you should expect to simply not get as much rent for a while. Be happy to get rent at all, IMO.
-W