In 2020, I supported myself with grants/ loans through college. I do not live with my parent. Can I be claimed as a dependent?

It depends. If grants or loans were your only sources of income, then you cannot file your taxes independently, simply because college grants and loans don't consider as taxable income and there is nothing to file. However, if you had other income, you can file independent if other requirements are met.  

You can learn more about that here:

Filing Requirements, Status, Dependents

You actually haven't provided enough info to answer your question. 

Did the parent co-sign the loan?  Which is more, scholarships or loans? What's the circumstances of "I do not live with my parent"? Being away at school, you still technically live with your parent. 

There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test. The support test is different for each type. The support test, for a QC, is only that the child didn't provide more than half his own support. The support test for a Qualifying Relative is that the taxpayer provided more than half the relative's support.

A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled
2. He did not provide more than 1/2 his own support. Scholarships are excluded from the support calculation
3. He lived with the parent (including temporary absences such as away at school) for more than half the year