I think that’s a fairly big “kicking the can down the road” assumption. Unrelated to OPs question, I see a lot of the “you can always go to masters at that Top school later” but that is not an option that is guaranteed or even available down the line. Decisions should be made based on what’s on hand right now.
OP - there are two distinct schools of thought. Some, like myself, will recommend splurging to go to a top school like CMU and others would recommend not paying that premium. There is no absolute right answer but I think you will benefit from drilling down even further on future goals.
For example, define what is a high paying job, who are the target companies, where do they typically hire from, how many alumns do you see from each school, is grad school even needed for those careers? etc.
You should really grill your parents on whether they can truly afford CMU? Will this mean they will have to take loans? Withdraw 401k etc..