Austin needs to ask for something less than a sleepover first and lead up to it instead so his dad will warm up to the idea of his son being close to Pete.
Have Pete over for dinner, or to hang out, and let his parents watch their interactions. Its not a bad idea for Austin to ask his mom to casually point out that Pete only acts out because his house is unstable, and that introducing him to more stable conditions (their own family) can be a good influence.
After that, he should ask his parents again and keep trying, inviting Pete over more and more often until the dad becomes more open to a larger compromise.