Colorado has a one month grace period for car registration renewals. As long as you get it registered in July, it shouldn't be a problem. (
Online Vehicle Registration Renewal FAQs)
Your car insurance should have all the info too. Your state might also have an online DMV system that already has all your info (my state does).
If you only have one set of keys, I'd just do it late and eat the fee rather than mail them if you can't get it done on time. My registration renewal was due in February and I forgot until I looked at my plates in May and went "oh crap." The only penalty here was a $10 late fee. (Maybe double check it wouldn't tank your parents insurance rates though, or that it's not an exorbitant amount in CO.)
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