I would say that it's much better to have their support going into the process. If you have a great school locally that you/they are generally happy with, and the notion of BS has not in their plans for you..."springing" the idea of BS on them on April 10, 2013 is most likely going to open a Pandora's Box of issues.
It certainly would for me.
This shouldn't be surprising. $120k for two years of HS is money that, even if they can afford that, perhaps they had other plans for.
Also note that, regardless of FA needs, I think most (all of the schools our daughter applied to did) require parents to fill out some statement.
Here's something else to consider...I'm sure BSs are more likely to admit students they know that the full support of parents upfront. Why would they admit someone if (since they are still minors), their parents are likely to put the kibosh on matriculation?
If you are searching for reasons to convince them, shouldn't these be the same reasons you felt it important to apply in the place? [These reasons are different for different students/families, so I won't try to offer any to you at this point.]