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When it comes to games and plots, there isn't really a way for us (as players) to get around the Flash problem, unfortunately. They're working on converting all the games and stuff to HTML5, but until that's done, there's simply a lot of stuff that's unavailable to players. (In case you're interested in reading more about this, JellyNeo has an article with more info about the end of Flash, and a list of playable games.)
I'm not sure if your question is about the (Flash) pet image no longer showing up on pet pages, but if it was: this page has a guide on how to get your pet's static image to show up on your petpage: http://www.neopets.com/~Shwoopsie.
And this page has some pre-made pet pages that you can use, which (as far as I can tell) don't use Flash for the pet image: https://www.neopets.com/~Xaia#Pd.
(Should you be interested in playing around with HMTL/CSS, online tutorials like W3Schools can be really useful. And if you ever need help with coding, you can always post in the Computers & Programming part of the TDN forums.)