I was lucky enough to have an elephant ride through the jungle in Thailand, can confirm that they are almost silent, only noise is the brush being pushed to one side on a path just wide enough for a person to walk. Amazing animals.
Not so lucky for the elephant though I'm afraid to say[1] :( The best way to experience elephants is on safari, where you can get close to truly wild elephants in their natural environment - something that to me at least feels much more rewarding than riding one in captivity. With a well chosen safari the fees you pay will be going to local guides and helping support and protect the animals too.
There are conservation parks, which to my surprise still let you do a limited amount of riding (as you wash them in the river) but are supposed to be rescuing elephants from mistreatment elsewhere. The riding is presumably offered as a way of funding the place (particularly given how much the elephants eat). So depending on where you go it may not be entirely terrible.
Have to admit I'm torn about the whole issue.
Animals in the wild are amazing but how may people can go see elephants in the wild. I recently brought my son to a dolphin show, while he has a new appreciation keeping intelligent animals in a tank that would normally roam over 1000s of KMs really bothers me.