I know a father can get an extension for 30 days after having a year's extension. Can he get another 30 days after that, or a couple of weeks?

As a parent of a Thai you can apply for a 60 day extension, but only once (per entry).

Currently it's also possible to apply for a 60 day Covid extension.

So you could apply for the Covid extension now, and then in 60 days for the parent extension (or another covid extension if it's still on offer then) for another 60 days.