NC as this is reasonably outing. We have experienced this twice, at primary (my school) and at secondary (DH's school). Overall, what I would say is that it's generally fine with staff, but can be tricky with parents. Primary was much harder for us in this regard, for two reasons - first, because relationships between parents are much closer at primary, and second because I was a parent at the school before I worked there, which meant I'd already developed friendships with the other parents. I won't go into detail as it's less relevant to your current situation, but suffice to say it became really quite awkward and claustrophobic, and contributed to me moving schools. (This was all about parent relationships. The relationship with staff was completely fine - I basically didn't talk about my kids at school, never saw them that much, and never sought any special treatment for them. If anything it brought a few benefits, like it being easier to nip into assemblies etc - only stuff that parents were invited to attend (again, strictly not seeking any special treatment), but things that I might otherwise have found it hard to get time off work for).
At secondary, it's been far easier. Partly this is because secondary schools are bigger, and parents have far, far less contact with other parents. DH has never taught either of the kids (only the odd cover lesson), though it's possible it could happen later in. They basically ignore each other when they do walk past each other at school. The other kids have gradually come to know that the kids' dad works at the school, but it hasn't been a problem for them. DH never goes to parent events except if he's there in an official capacity - he wants to have no personal relationship with other parents whatsoever. If there are parent events we want to go to, then I go alone - I'm able to be perfectly friendly and chatty with other parents, but I'm always very aware that I'm quasi-staff, so always remain a bit careful what I say. With other teachers, DH has never found it a problem - his colleagues might occasionally make a brief mention of one of the kids (normally in a slightly amusing anecdote kind of way), but it never really goes beyond that. As well as the parent social events, I usually go to parents' evenings alone, as DH feels that's less awkward for the other teachers - though sometimes he has come too and it's been fine - he's very good at switching between parent and colleague mode. He tries to be scrupulous about not looking up the kids' marks etc, although occasionally he has to see things as part of data reviews etc.
So, I would say it's perfectly doable at secondary, but you need to be scrupulous. If your colleagues start to over-share stuff about your son, I would politely rebuff it in a way that makes it clear that you don't want to be a parent while you're at work. With parents, avoid any WA groups etc like the plague. Do you have a DP who can do the 'parent bit' instead of you? That has made things far easier for us, as 'we' can still feel involved in what's going on, without feeling like we're blurring the boundaries.