Based on the following, I’m not entirely sure that’s the case.
The family is probably willing to “stretch” further for a more prestigious school than a non-prestigious school, but they may be of the generation when “prestigious” schools still gave merit aid or they read news articles about someone getting a “full ride” to a Top X school, not realizing that the “full ride” was need-based, not merit-based. Depending on how far north of $200k the family’s income is, paying approximately $1.6 million for college ($400k/kid for 4 kids to a Top X school) could very well be out of the willing/able range for the family.
Alternatively, the parents know what the financial situation is like at college nowadays but don’t feel like telling their kid not to go for it. But when OP gets in but can’t fund the other half of the costs, the parents say, well, then you can’t go, when it was never a possibility anyway. I suspect it’s the former situation more than the latter, though.
As the parents have 4 kids to put through college, I personally suspect that the parents are going to say that they’ll pay the equivalent of UIUC’s cost of attendance and if OP wants to go anywhere else, merit aid, loans, and work will be required. But I am not OP’s parents, so that’s just a guess on my part.