It depends. You are correct, if you do not submit your 1095A with your tax return it will be rejected if you are not a dependent. If you were on your parents plan, both you and them will need to enter the 1095-A. If you are a dependent, then yes, you can submit your return without the 1095-A.
If this was a shared policy, you will both need to enter the form on your returns. You will need to get the form from your parents and enter it exactly as it appears. After the form is entered you will be asked if this was a shared policy, if it was you will click yes. You will also select the months it was shared for. Then you will be able to allocate the amount to you and your parents. You can claim 100% or 0%, they can do the same. Many times parents paid the premiums and will allocate the full amount to themselves, while the kids just enter the form and allocate 0%
It is not likely that your parents need to amend their return if it was accepted. If they had the policy, then they would have had to include the 1095A on their return or it would have been rejected.
If this was your policy and they were not a part of it, then they cannot include it on their return.
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