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:
https://ttlc.intuit.com/community/taxes/discussion/can-i-claim-my-daughter-who-got-married-on-decemb...
