It's not related to your browser/system at all: Flash has been discontinued - not by Neopets, but in general. It's dead.
(Adobe actually recommends uninstalling Flash entirely, as it's no longer supported, which can lead to security issues. As you've noticed @Eagle_eye129, downloading older versions of Flash can even result in infecting your computer/phone with a virus.)
Neopets has been working on converting everything to HTML5, but it's taking a long time. This article on JellyNeo has more info about the death of Flash, and which things have already been converted/still need to get done.