Can a student claim their apartment rent while at university in ontario if they live with their parents in the summer and their parents address is their mailing address?

Yes you can apply for Trillium Benefit (tenant declaration) for the months you occupied and paid rent for your rental in Ontario during 2018. Remember to stipulate whether you need monthly payout of the benefit. As well as the OEPTC component of Trilium, you will receive the OST component.

There is no tax credit that you can claim for rental per se. At least not innOntario.