Try running the net price calculators on at least a few LACs that claim to meet 100% of demonstrated need. The following list of ~60 colleges includes many LACs:
https://www.usnews.com/education/best-colleges/paying-for-college/articles/2016-09-19/colleges-that-claim-to-meet-full-financial-need
If your parents cannot cover the EFCs for any schools on that list that are realistic for admission, then you might want to look for LACs that award relatively large/numerous competitive merit scholarships. Check the Kiplinger rankings for that information:
http://www.kiplinger.com/tool/college/T014-S001-kiplinger-s-best-values-in-private-colleges/index.php?table=lib_arts
Especially if it looks like none of those will be affordable, even with above-average merit grants, also have a look at colleges that guarantee large merit scholarships for qualifying stats:
http://automaticfulltuition.yolasite.com/