I can relate; I cannot tune out noise either (never have been able to), and it does add to my stress level. People noise is especially distressing to me, though particularly when I'm trying to get something done that requires (for me) relative quiet.
Is it acceptable for me to tell a parent that the noises their children make are bothering me? And if so, what is the best way to inform them without making the situation worse?
I think that depends on the type of noise the child is making; is it really outside of socially acceptable norms? If it truly is, then it's acceptable. If it's not, then it's not.
One major difference between my experience and yours is that I only ever perceived this as my problem. I never considered that I had any right to ask people to stop making noise for my sake. (Please don't get me wrong; I have often fervently wished that someone would just stop talking!)
I guess what I'm saying is that the attitude with which you are approaching this affects the way you might seek to deal with this problem. If you approach this with an attitude that you have a right to/need to control the noise level around you, you will desire to trespass on the rights of others. If you approach this with an attitude that you have a problem with noise level around you - which is not the responsibility of others to fix - you will seek solutions by dealing with your own levels of anxiety/irritation/other that will give you ways to soothe yourself in all such situations.
...the child just couldn't stay quiet: she was singing constantly, saying "byebye, please come in" to every car that we came across, even parked cars, being very annoying in general.
Maybe it was just excess energy or ebullience; maybe the child had a neurological disorder of her own. In any case, it is actually her right to sing and speak. I cannot myself imagine asking someone on public transportation to quiet their child. Most parents try to balance the needs of others against the needs of the child - that's commonly known as making their child behave properly in public. If the parent isn't doing something in that situation, either they don't know it's a problem or don't care about you - in which case, they will not take kindly to your request - or they know there's a problem, and they have decided that the only reasonable way to deal with it is to allow the child to chatter on.
In any case, I see a problem with the fundamental way you're approaching this.
Please consider that you might be asking people to give up freedoms they have in order to help you feel better (something that is not looked favorably upon in general). Playing loud music in the middle of the night? Your need wins. Dog constantly barking in the house next door? Check. You win that one, too. Kids running up and down the stairs all day in the apartment adjoining yours? Check. Evangelicals preaching to you at your front door? Check. Telemarketers bothering you? Check. Child singing and chattering on a bus? NOPE.
Work on developing coping skills for such situations. They will help you far more than the alternative.
Such skills can be to meditate (with practice you can do it almost anywhere), to practice mindfulness (paying attention in a particular way: on purpose, in the present moment, non-judgmentally), to play games in your mind or on a phablet, to carry noise-cancelling earphones wherever you go, to listen to music that you especially like through earphones; to imagine what it is like to be that little girl in her sing-song world, making up a non-judgmental different reality, to practice empathy. Any of a vast number of strategies can be learned to deal with stress.
Coping with stress: Divergent strategies of optimists and pessimists.