You're right in that it makes no logical sense so my best guess is that it's an emotional thing for them they have to work through.  

Since they are not pets but are needed for medical therapy and emotional therapy dogs.

@Mila F. top of my least favorite tenant list, I suddenly need emotional support. 

Way better than having g and ESA emotional support animal.

Especially when so many people hopping on here want to give the exact same emotionally-charged, offended, and "personal" opinion over and over...and over.

The husband is a pretty emotional person, he kept saying that all the memories of his kids are in that house, and he would not be able to sell the place!!

It is often too difficult emotionally or mentally exhausting to give up their previous home.

Time is running out Wife and 3 Children LIVE in the home and want to CONTINUE to live in the home as the Children are entering or in High School(a painful time to have to move or relocate for all) - YES lots of emotions all around involved.

My theory is it's much easier, emotionally, to walk away from a rental than it is to sell your home (home owners are more likely in for the long haul).

If they inherited the home, there are likely emotional bonds they have with the property.