Talk to the other parents, see if they are allowed to stay - or to drop in at any time they wish. If not - don't go there.
Look into how the school treats women and girls, there are schools out there that teach kids that women and girls are "less" than men and do not have to be listened to - even mom and teachers.
Listen, carefully, how the students interact with each other and teachers and how the teachers talk to students - this will tell you a lot about the etiquette in the school.
In my school kids and adult students do not have classes at the same and while adults do sometimes make "guest teacher" appearances kids do not have the right to correct adults. The reason being is that kids bodies and adult bodies are not the same and sometimes forms, moves and other things may be changed from what the kids learn to suit the adults. Also most adults in our school have been taught both the kid's and adult's versions so that they can become teachers....the kids don't have both versions, hence the restriction.
beware of a school that won't let a prospective client view group classes in all age groups,,,,doing so can be eye opening as what to expect as your kid progresses before you hit that level. Private lessons are ok to say no to viewing.
Check to see what expressions of respect the students and teachers give each other, and the school. This will vary depending on the culture of the school and if you are not comfortable with any of it - don't go there.
The first three are real deal breakers for me....and would send me running from any school that made me uncomfortable on any of them.