When transferring a tuition amount to my parent, do I transfer the exact amount that they owe to CRA?

Not quite; you can transfer up to $5,000 to your parent, depending on the tax calculations. 

(For more information, check out: https://www.canada.ca/en/revenue-agency/services/tax/individuals/topics/about-your-tax-return/tax-return/completing-a-tax-return/deductions-credits-expenses/line-32300-your-tuition-education-textbook-amounts/transferring-carrying-forward-amounts.html)