Logically, there's no reason why not, but it sounds strange to me.
So normally pet peeves are about other people's shortcomings?
[AmE - not a teacher]
I would say yes, always about someone else. Also, pet peeves are supposed to be with issues that most others do not find annoying.
Aren't there common pet peeves, which many people don't like? For example, pet peeves like people who don't answer phone calls, or don't reply to emails, dogs barking late at night, or people who jump queues? Or aren't these things "pet peeves"?
A perusal of 'pet peeve' at OneLook Dictionary Search seems to confirm that all of those cited above by birgit33 fit the definition.
Perhaps you could provide a citation, because the first 8 links I got on onelook matched none of those examples (and indeed, the examples at wordnik were rather specific to the speaker). In fact, one of the links was to wikipedia which seems to bolster my opinion: "A pet peeve (or pet hate) is a minor annoyance that an individual identifies as particularly annoying to him or her, to a greater degree than others may find it." (emphasis mine) Pet peeve - Wikipedia, the free encyclopedia
Do you consider Wikipedia the last word on everything? I can't see how you can dismiss all the examples in post #5. Your interpretation of the OneLook citations seems to be different from mine. The fact that a pet peeve may be a personal dislike does not necessarily mean that others do not find it annoying. One of my pet peeves is people who rearrange their hair at the table in restaurants. While some may tolerate this activity, others may not. IMO, it certainly qualifies as a pet peeve.