If you can be claimed as a dependent, you must check the box that says you can be claimed as a dependent, and you are disqualified from making any HSA contributions. You can be covered by your parent's insurance while not being a tax dependent, because the age limit for being a tax dependent is 18 or 23 and the limit for being covered by a parents insurance is 26.
First make sure you can't be claimed as a tax dependent. You can be claimed as a tax dependent if you are age 23 or under and also a full time student, and if you provided less than half your own support over the whole year. (You were living alone for 3 months but supported by your parents for 9 months.) If you are age 24 or older, or age 19 or older and not a full time student, you can't be claimed as a tax dependent unless you are disabled.
Then, ACA insurance can be eligible for HSA contributions (can be an eligible HDHP), it depends on the coverage limits.
For a family insurance plan, it must have a deductible equal or higher than $2700, and the annual out of pocket maximum expense must be $13,500 or lower. If you aren't sure, ask your parents or check the web site for their insurance plan.
If you were covered by a qualifying family plan, then you would check "family plan" for those months, even though it was in your parents name, and then "Self only" plan for the months where you had you own insurance (but, only check "Self" if you have self coverage on the 1st of the month. If your job and insurance started September 15, you check "self" starting for October.)
If your parents insurance was not qualified, then you were covered by non-qualified insurance for those months and self insurance for the end of the year.