When does Tricare Pay after Medicare ?

Once you are eligible for Medicare, Tricare then becomes a gap policy. Everything is filed through Medicare first, then the hospital or doc will refile the remaining with Tricare. Anything left... is on you.

The last few hospital stays, my parents (retired AF) have only been out of pocket about $1000 for each one.

Most people who are entitled to Part A under Medicare, have to pay for Part B in order to get Tricare. Tricare does not pay first for services covered by Medicare. Tricare pays after Medicare has paid.