Can a parent carry forward the tuition amount that has been transferred to them for following years income tax if they do not use it in this years return?

As per the Canada Revenue Agency(CRA), you will not able to transfer the tuition amount to anyone if you carry forward. Please refer to this CRA link for more detail: 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

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