I can only access the colour pallet of the brush in the menu on the right, and not in the header menu bar. Has it been removed from the top all together?
That seems indeed to be removed from the top bar tool settings. Don't know why. I'll try and ask Pablo Vazquez on Friday.
Yikes I hope they don't remove that. I still see it with Blender 2.82a. Which version are you using jjaydens?
That is utterly strange, Kent. I can't find it in 2.82a (or 2.83 for that matter). It used to be under Brush. (I looked in the other places as well of course.)
This is 2.82a on Linux...From the official website.
spikeyxxx I apologize, I thought we were talking about the single color swatch:
Sorry for the confusion. I can confirm that the full color palette UI is missing from the Brush menu in 2.82a.
The work around I found was to enable the color palette add on in the preferences. The new palette is located in the right side "N" menu. 
If you want to restore this functionality in version 2.82: under Edit -> Preferences... there is an add-on called Paint Palettes.
This adds a new tool to your N menu (top right of your main view) called 'Paint'. 
To add a new palette, click the '+' next to Index. To save the palette you've created, click the '+' next to where the word 'Wood' is in the above screenshot. It works pretty much identically to shown in the video, except you have to eyedropper it into your current palette..
ddavide789 thank you, but you are a month behind ndtith11 ;)
Anyway, don't think that the Color Palette has been removed from Blender, it just isn't in the Header anymore.
You can still access it in the Tool Settings in the Properties Editor:
But when you go Fullscreen on the 3DViewport when painting, the addon can be useful!