Ofsted, on receiving such a complaint, would either:
a) Ignore it if it is simply ranting
b) Suggest a parent use the school complaint procedure or direct them appropriately.
c) If they had any concerns, would, through their regional link, speak to the local authority/regional schools commissioner office links for relevant area and discuss the issue and relevant school to see if any concerns were in their sightlines, needed follow-up, were already being monitored
Ofsted regularly raise issues reported to them by parents with us (LA) in their regular meetings with us about our schools. Mostly to tell us there has been contact. Most concerns raised by parents are personal to them and they are annoyed because the concern has not reached the outcome they want eg a child/children excluded, a member of staff disciplined, their own child not excluded. That is not an issue for Ofsted. Most concerns are already known to us (LA) because they have also been reported to us by the parent and the school and we have already advised on action/outcome and are satisfied it is being appropriately dealt with, or we are already supporting and monitoring the school. Occasionally, OFSTED will ask us to look into something and report back to them where we feel the school is on a matter - an example would be they recently asked us to look (in a particular secondary school) at the attitudes of older teenage boys (Y10-11) towards girls in a school, whether the school was doing any work on this and how any incidents had been dealt with and outcomes followed-up. It was quite an unusual request. The school was dealing with several incidents, had a programme in place with a group of boys, was monitoring the effect of it and the boys' behaviours carefully and doing student voice work with girls about how they felt they are treated. It was an issue, had been recognised, appropriate action put in place and outcomes were being carefully monitored and the outcomes were positive so far. That is all anyone could ask.
d) Rarely, it might raise a school's profile with them if we are unable to provide reassurance and there are already concerns about a school - which could put them into an early inspection slot than we would have expected.
e) Very rarely they could decide to inspect a school immediately - likely on a Safeguarding basis. I have not known this happen but it does.
I think parents who shout and threaten'I'm going to Ofsted' think Ofsted jump at their phone call or that the Head is terrified. I imagine every Head hears it from angry parents quite regularly. I did as a Deputy Head and I do as a School Improvement Advisor for a local authority. Probably the first couple of times as Deputy Head I felt a but worried but I wouldn't now. I know we do the right things- not doing what a parent wants is not an Ofsted matter.
However, it is important that the route is available to parents- there are occasions where it is needed.
If Ofsted had a very serious matter about a school raised with them, that a LA was unaware of , and once brought to our attention we were concerned by and unable to provide reassurance that it was already being addressed, I would think it reasonable that it prompted an inspection. It has not happened.
Local authorities and RSCs Offices talk to Ofsted regularly and share info about all schools. There should be no surprises if the system works well.