There's a site on the web with a sort of basic trailer with a platform with 2 bike wheels but you can build them too.

Oh, here's an idea, see if you can score a decent wheelchair, like a Quickie. Then, attach a bar to it that attaches to your seat post. Make the wheelchair tilt back, it's up to you and your dad whether he wants to ride looking forward or looking back. That makes it a lot more stable, since by sitting back you'll get a "bucket seat" effect tending to keep the passenger in.

Most wheelchair users especially nowadays weigh more than 150 lbs so modern chairs are built sturdy enough, and I haven't priced them but they're probably not too much.