Most colleges have an online Net Price Calculator, it will give you a rough estimate. Go to each college that your kid is interested in applying, find the net cost. If you are confident that you can afford without outside aid, then you most likely do not need to apply for aid.
The college does not care if you apply or not. Though it will definitely increase the chance of admission if you apply for aid, but in fact you don't need aid. This is especially true for need-aware admissions.
If you need aid, do not game the system for admission. There is no point of applying for admission if you can't afford it without sufficient aid.
In the end, it is up to the parents to decide.