Can a full time college student that is claimed on their parents return also take the standard deductions for their taxes. I don't see an option not to take the standard.

Yes, you get a standard deduction even if you are claimed on your parent's return.   It is applied automatically, you don't need to select it.    Look on Line 12 on Form 1040.

If you are being claimed as a dependent, your Standard Deduction is lower.  For example, in 2023, the Standard Deduction for dependents is whichever is greater:

1. $1,250
2. Earned income plus $400

Note that for option two, it can't equal more than the Standard Deduction for your filing status. So, if you are a single taxpayer under age 65, it couldn't exceed the $13,850 Standard Deduction for 2023.

But your Standard Deduction might be limited.

The Standard Deduction for an individual for whom an exemption can be claimed on another person's tax return is limited to the greater of:

…..$1,250 or

…..Your earned income for the year + $400 (but not more than 13,850))

So you would get a minimum of $1,250.

See Table 8 Standard Deduction worksheet for Dependents on Page 26

https://www.irs.gov/pub/irs-pdf/p501.pdf

Or the 1040 worksheet on page 34

https://www.irs.gov/pub/irs-pdf/i1040gi.pdf