Galaxy Tab 3

Have you tried the Youtube app? That doesn't require Flash. And the mobile version of Youtube should play in HTML5 anyway. Are you using the default browser?
 
It does, but Adobe pulled the app from the Play store ages ago. They have been updating it on their site to fix security holes, but that has been stopped last September. You can still grab the apk of it here: http://helpx.adobe.com/flash-player/kb/archived-flash-player-versions.html

Scroll down or Ctrl+F "Flash Player for Android 4.0 archives" and the latest version is 11.1.115.81.

However, this will also depend on your tablet version. The Flash app will work fine with 7.0" and the 8.0" model, but the 10.1" model uses an Intel SoC, and the app is not compatible with the x86 ISA, only ARM. Also, if you do ever get the Android 4.4 update, Flash won't work with this either, as Google has changed things with the OS that prevents Flash from working.
 
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