I am married but a dependent (student under 24, lived by parents, supported by parents). Can I qualify for the Earned Income Credit by filing Married Filing Jointly?

No.  A person, without children, must be 25 or older, to get EIC, regardless of marital status.

However, if your spouse is over 24, you would have to file jointly to enable him/her to claim EIC.  Your joint earned income would be used to calculate the EIC, even though you are under 25.  

So, yes, maybe. If your spouse is 25+, and your joint income is on the up slope of the EIC curve, you will effectively get EIC , for your income, by filing a joint return.  If your joint income is on the down slope of the EIC curve, your income will reduce your spouse's theoretical EIC.  But that's academic; he cannot get EIC filing separately.

But if you use Married Filing Jointly (MFJ) filing status, your parents can not claim you as a dependent.

For more about how married dependents should file, see 

https://ttlc.intuit.com/questions/2938663-can-i-claim-my-daughter-who-got-married-on-december-19th-2015?jump_to=similar-questions

Basically, it's YOUR choice, not your parents, whether you are claimed as a dependent OR you file as Married Filing Jointly. The best thing to do is for both couples to prepare their taxes both ways and see how the family comes out best overall, before filing.