Can I file a car registration cost if my parents own my car?

No.  If you are not the car owner then you are not legally liable to pay the registration.

Car registration is not deductible, only the *part* of registration that is a property tax based on the value of the vehicle is deductible and not all states have such a tax.

That would be an itemized deduction that would do no good unless ALL you itemized deductions were MORE than your standard deduction.   If you are filing single then unless you have deductions MORE than $12,200 there is no benefit.

2019 standard deductions

$12,200 Single
$18,350 Head of Household
$24,400 Married Jointly

Add an additional $1,300 for over age 65 or blind
This amount increases to $1,650 if the taxpayer is also unmarried.