This is a gap in the law, don't know if it is intentional or not.
If you are covered by an HDHP, you can contribute to an HSA in your own name, and you are not subject any limitations based on your parents's contributions. In other words, if your father has a family HDHP that covers you, your father and mother can each contribute $7200 (for 2021) or $7300 (for 2022), but their maximum combined family contribution is also $7200 or $7300. If you are covered by the same family HDHP, you have a separate contribution limit of of $7200 or $7300 and your limit is not reduced by their contributions.
If you were eligible in 2021, you can make retroactive contributions for 2021 until April 18, 2022. (However, don't wait to the last minute, as the bank will likely take a few days to process the deposit.). Many banks offer private HSAs, since you can't use your parents' workplace HSA. Shop around to get the best terms (monthly maintenance fees, minimum deposit, and so on.) If you plan to make a retroactive 2021 contribution, make sure the bank knows in advance—there will be a form or a selection on the web site—so it doesn't go in for 2022 by mistake.
