I know you've been through a lot lately, so I hope my response to this doesn't sound unsupportive, but here's my honest take on what the problem is:
"they wanted me to move their child to the 4 year old class because next year she will be attending kindergarten in a VERY expensive private school. I told them she misses the cut off date of Oct 1 and she was doing great in her class and didn't see a need to move her"
The above paragraph is actually where I think your issue lies (or at least begins) - not with the babysitter.
These parents are planning on putting their child in kindergarten next year. Consequently, I understand their desire to have her in a Pre-K class with four year olds who will also be making that move next year (not with three year olds) . You are sticking to an Oct 1 deadline which is obviously not a deadline that their Kindergarten will be using. If the child is unable to keep up with the older class, or does not have the maturity level that the other children have, then that is something the parents need to see and experience and understand for their child.
But your choice is to keep her in a class with three year olds, and then send her off to kindergarten next year (where she will be suddenly placed with all older children). I can see where the parents would have a problem with this.
I don't necessarily agree with an Oct 1st child starting kindergarten before turning 5, but since that is the parent's intent, it seems you are not supporting that. Even if the curriculum is sufficient -- the child is not getting the social interaction with older children that she will need to be prepared to function successfully in kindergarten.
I can also understand why the parents , given that they are frustrated and confused by your placement of their child this year -- might want to find out more about the school/curriculum from the babysitter/teacher who happens to be there in their house. She isn't their child's teacher, but she is a teacher at your school. For them to ask her questions about the school,classes etc, is just normal.
And again, I hope you don't think this sounds too harsh, but I just think your response to the parents sounds controlling and reprimanding: " I expressed to the parents that I have no problem with them speaking to their child's teacher, or myself, however I do have issue with them asking staff such pertinent questions.
i think it's perfectly acceptable for you to have discussions with your staff about boundaries and about being circumspect in their off campus conversations (especially since they sound like a gossipy group) -- but I think it's just not going to go over well if you tell parents they can't engage in a conversation about school/classes/curriculum with a teacher in their own home.
Bottom line, I think these parents will be taking their children out of your school not to be with a nanny full time -- but to be in a pre-school program that they feel works in tandem with them to best prepare their child for kindergarten.
I think they probably feel that you have made a decision for their child's placement this year that is not acknowledging of their needs and plans. I know you offered to revisit midyear , but that could be perceived as more disruptive than helpful -- as it doesn't give the child the kind of continuity she will need the year before kindergarten. I know I wouldn't want my child uprooted mid-year and put into a new class where she'd have to adjust to new friends, and a new teacher prior to doing that all over again in kindergarten.
I do think you will want to change your policy about staff members babysitting because it blurs lines and does allow for a gossip mill to churn. But if the parents didn't have the pressure of Kindergarten and the basic conflict with you over how their child's last preschool year unfolds -- I don't think the situation would be what it is today.
As a preschool director, your judgement calls about what a child needs and their maturity level I'm sure are well informed I would also not recommend kindergarten for a child with an Oct 1st birthday who will not be 5 when starting -- but given that is their goal, I just think the parents felt you were working at odds with them and that's the root of this problem.
But I'd certainly watch that gossipy employee as she seems quite happy taking snipes at everybody.
Good luck with this. Parents get very crazy the year before Kindergarten, especially if they are applying to exclusive schools. You want to make sure they feel you are on their side and working with them