They could seeing you as the person that 'picked on' them as a child, not as their child's teacher.
If they felt that a teacher had it in for them and made them miserable at school, when their kid moans that Miss PC gave them detention when they're weren't even doing anything, they could assume that
a) Their child is upset and they don't want their baby to be upset
b) Their child is always lovely at home and could never be a complete little shit or bone idle brat for anybody who was nice to them
c) That all teachers are the same as the one that gave them detention or made them cry aged 13.
What sort of things are they complaining about? Behaviour Management? Sanctions? Grades?
It could be that you do have things you could work on - it could be that you are being subconsciously harsher on them because you know they don't have the same degree of deprivation and disadvantage that your previous students did, or that you react strongly to argumentative little wotsits because they're so much better at digging at you subtly over hours.
Or it could be simply that they just haven't realised a lot of lovely children undergo a complete personality change once they travel in packs through the school gate and have to shut up, sit down and do as they are told; some of the kids with the worst behaviour records for rudeness/disruption in class can be absolutely lovely to the staff that they don't see as making them do things they don't want to do - and also behave brilliantly at home.
I found moving from an Inadequate school with huge levels of deprivation was a completely different world to the Outstanding, much higher average income, fewer single parent families, almost no PPG student, school.
For a start, I was having to deal with children who would simply cry at me anytime I said 'No' and expected to go home as soon as they said their tummy hurt or they had a bruise on their knee. Mum was always able to leave work (if she worked at all) and be there in the car within twenty minutes or Dad could take the rest of the week off to look after them - no telling me there was no point calling because she wasn't able to take calls/couldn't afford to leave early/might lose her job and, in any case, they had their siblings to pick up from Primary and look after until 8pm - and, of course, their parents were, having given them that nice, safe, gentle life, extremely defensive of them if they thought they were being treated harshly or unkindly.
In comparison, with the kids at the Inadequate school, somebody speaking to them firmly but not unpleasantly was probably the kindest things they'd heard all week. So they wouldn't be complaining.
I also think that the parents of your new school are probably very good at tapping into exactly the right words to get to you, as maybe you come from a similar background? Somebody threatening to come up the school and Sort You Out is horrible, but it's alien and obviously wrong, whereas the slight derision and change in tone to make you feel shit is exactly what hurt you when you were a child You don't have the same level of defence against that sort of bollocks.
Whatever way it is, if you've looked, reflected and decided that it is unfair and you are doing things right, you would be entirely reasonable to work on pretending/letting them think their words have absolutely zero effect upon you.