If the parents all say the same thing, then you will have social proof, but it will be wrong.
You must, regardless of the hurdles, find a way to survey children or graduated children. When parents find out that their child was not socialized, despite their strong belief to the contrary, they will be left with children who no longer wish to be around them.
You can easily design a study to steer people to homeschooling, by only asking the parents. They have too much invested in being alternative education experts and it's hard to admit that you played around and messed up with how you schooled your children. If homeschooling turn out ok, the upside is that a child will have a maybe slightly better educational experience. If homeschooling turns out to not be ok, the downside is that you will permanently damage parent-child relationships.
You're playing with fire by even attempting this study, the way you have it set up right now.
Oh boy. Before anyone goes too far believing this paid consultant, you should really be surveying homeschooled children who are now adults.
I say that because I'm a homeschooled child who is now an adult. From my immersion and circle of friends who were also homeschooled, these are my takeaways:
Parents are untrustworthy proxies for their children's experiences. If you surveyed parents, they would all say that they kept their child socialized through homeschool groups. If you surveyed the children, they would strongly disagree. The point made in the ask HN "and have witnessed it to be very social" is the same one every homeschooling parent I've ever met makes, and pretty much every homeschooling child disagrees with. (And I've met a lot of homeschooling parents!)
An astonishing number of homeschooled acquaintances I know have estranged themselves from their parents, and would cite their decision to homeschool as a factor in that decision (> 50%, n=~25.) Of the 5 children in my family, 4 cite homeschooling as a lingering negative that affects their social relationships to this day. 3 have forgiven my parents for it. The others still limit contact with them. (Siblings are all around their 30s.)
Children who are still school aged are incredibly unwilling to insinuate that homeschooling doesn't work for fear of the consequences. If you see your parents devote their time to homeschooling you and are constantly saying stuff like "You'd think that homeschooling isn't social, but look! We found 5 other weird families that our children only pretend to like because they're desperate for a peer group, and they get along OK!" you would also be hesitant to say to your parents that they approach they've dedicated their life to makes you lonely.
This is a decision, that, if you mess it up, you really mess it up. Be cautious interpreting a study that only asks parents their opinions, given by a consultant with a self-admitted homeschooling bias.
You must, regardless of the hurdles, find a way to survey children or graduated children. When parents find out that their child was not socialized, despite their strong belief to the contrary, they will be left with children who no longer wish to be around them.
You can easily design a study to steer people to homeschooling, by only asking the parents. They have too much invested in being alternative education experts and it's hard to admit that you played around and messed up with how you schooled your children. If homeschooling turn out ok, the upside is that a child will have a maybe slightly better educational experience. If homeschooling turns out to not be ok, the downside is that you will permanently damage parent-child relationships.
You're playing with fire by even attempting this study, the way you have it set up right now.