I feel for the OP, but unfortunately I am not sure what they would like BA to do. I've also been in F a number of times with toddlers in the cabin and each time parents have generally been excellent at managing their kids. Sure, you get some noise on occasion, usually on takeoff and landing, I assume this correlates to air pressure changes, but typically any noise is limited in terms of time. Yes, invest in noise cancellers, I used to use ear plugs plus over ear Bose, I now use in-ear Bose noise cancellers plus the airline supplied over ear noise cancelling headset.

This does remind me of a return flight in J a few years ago LHR-NRT-LHR.

On the outbound were a Japanese couple with their young baby who clearly had a bad skin complaint (poor tot was continuously bleeding), and they went above and beyond to placate the poor chap. Of course, this was far from a noise-free flight, but I was deeply appreciative of the level of attention paid by the parents, as well as the crew's help, dealing with the child's distress.

In contrast, on the return a couple of days later, a British couple with two children aged about three and five, did absolutely nothing to control their two kids. It was left to crew to control them. The kids refused remain in their seats for takeoff and landing, and the parents, despite sitting adjacent to their children, left the crew to attempt to control them throughout the flight. I was left to conclude that they were used to having a full time child minder, and were incapable of parenting themselves. At immigration I found myself directly behind that British family. It was all I could do not to say something I might later regret, but my anger at their complete indifference and lack of responsibility meant I had to bite my lip very hard.

I did once respectfully ask a parent to please "be careful" with their child in the BA lounge in LGW who was running around out of control throughout the lounge after I nearly tripped over them with a hot cup of coffee. Rather than being met with apology and embarrassment, I was met a contemptuous and angry response: I left for fear that he might become violent.

Crew are not parents' child minders, and it is reasonable to expect anyone who's a parent to understand their own responsibilities, and by far most parents do indeed do so to their eternal credit. A small minority of parents and guardians do live in a selfish bubble apparently bereft of self-awareness, and regrettably occasionally we all have to deal with the consequences whether on a plane or not.

Sometimes you have to deal with something the parent may not be aware of, which is never easy, it is better to do it calmly early on before you are completely beside yourself. Sometimes the parents may not even be aware of it, such as seat kicking.