You definitely need "for." In addition, it's best to avoid using the passive voice unnecessarily. The only reasons to use the passive voice would be if you don't know who paid for it, or if it's obvious or unimportant who did so. I suggest that you use the active voice and make who paid for it the subject of the sentence: My dad paid for the dinner. Also, if you change the object, you might not need for: My dad paid the bill. My dad paid the check. My dad paid the tab.

The dinner was paid for ... followed by an explanation. The dinner was paid for by my brother. The dinner was paid for in advance. The dinner has been paid for already. The dinner was paid .. is an incomplete sentence.

Paid for