I love to hear that your parents want to come and stay for so long to visit your daughter. My father just moved down here in May because he wanted to be with us and our son (his only grandkid). I wanted to respond because my dad is doing a little handyman work on the side to keep himself busy (there are only so many chores to do around our house!). He's been in the construction business since before I was born so a little over 30 years. Back home, he did A LOT of handyman work for elderly friends and family of his grandmother. His name is Dean and his cell phone number is ###-###-####.
Hope you're able to find somewhere for your parents to stay when they come. Aside from extended stay places, I am not sure of other options. There is company called Home Away that rents vacation rentals. They are based out of Austin so I bet they have a handful of property listed in Austin. They might be pricey though. Good luck!
--B.