There are a lot of factors we do not know because we do not know your parents side of the story.
The Prior Obligation alone is enough reason but for the sake of discussion lets add a few more possibilities:
Will there be a RESPONSIBLE adult with you at all times
Where will you sleep/ eat -Will there be costs involved
Are you a strong swimmer
Do you burn easily and forget to put on block
Have you always been responsible and stayed out of trouble
Has your friend ever been in trouble
Do you have any allergies or health conditions that could be life threatening
Is the beach area safe IE: Gang activity or rip tides
Aside of the above questions:
If you have a prior obligation (Church event) that trumps spontaneous trip to beach.
Your parents do not sound psychotic- they are some of the few that CARE enough to guide you to a solid adulthood. They have good reason for not allowing you a solo trip with a buddy for a week.