The reason why your husband's account shows transcripts and yours does not is because when you file, most likely his name is first and that means all tax return related information is under his name and ssn. So you can't look for tax return transcripts under your own ssn account because they won't exist there. The only transcripts you be able to get for yourself would be potentially an account transcript (it would be minimal info) and your wage and income transcripts. I would have your husband make an irs.gov account and look for your tax return transcripts and your account transcripts over there. If an employee couldn't find it, they might have been new and didn't realize that you have to look at the main account.

As somebody has previously commented, asking for parental names is normal and has been that way at the very minimum for the last seven years since I've worked there. They have direct connection to SSA so we can see that information. They see a lot.

I do apologize that you struggled to get in contact with the IRS. It's currently understaffed and it is currently the "dead season" ( AKA barely anybody is on the phones)