I was in school for both the spring and summer of 2019 and living as a dependent with my parents.
I am and was employed but was still on their benefits plan.
My husband and I got married in early October and are now trying to figure out what to do about our taxes.
Do we have to file married but separately since both are parents will be claiming us for a part of the year?
There is no such thing as claiming dependents for a part of the year. If your parents are claimng you for the year, then yes, you both must use Married Filing Separately (MFS) filing status.
As to whether they are allowed to claim you and should claim you, see this similar question:
It doesn't matter who paid for school. If you decide that it is better to claim yourselves and file as Married Filing Jointly (MFJ), you will be allowed to claim the tuition credit, on your tax return, even though your parents paid your tuition!