Instead of focusing on palette inspiration, one should aim for art inspiration. What do I want to create? What do I want to achieve with my creation?
This being said, there are times when an artwork or photo has interesting colors you might want to explore further. This is a great way to look at how colors interact and how to implement them.
Essentially it's the artwork that dictates the palette's quality.
Making such posts won't improve anything about that. There's no single valid method of making art or palettes, but one thing that can be definitely improved is self-esteem and will to learn. Please post your palettes in the palette Feedback thread and see if people can help you!
Personally palette creation is highly defined by the context you're using it in. Therefore I create palettes as I create a piece - it's never set in stone. This doesn't mean my process is random, however.
First of all I use colors that are close enough to the feel I want. This serves to get a feel of the colors as I make the art. References are paramount in this step, since sometimes the colors might react in an "unusual" way, such as when close to strong, colored lightsources.
As I finish blocking out the colors I see which one of them are superfluous to the piece, by replacing them by already existing colors. Tweaking the colors so that they can be used in several contexts in the piece, checking values and contrast, and so on. In this step I analyze the shades and decide which hues require less contrast, and which doesn't. Remember that sometimes low contrast can be useful in pixelart to suggest very soft and smooth surface, or misty landscapes. This is, too, an example of context making the colors feel useful.
This approach feels better to me because the palette will be tailor-made to the piece I'm making. Lastly I run the finished image into contrast filters to zero in on a good balance (sometimes, manual adjustments only get me so far due to being accustomed to the piece. Adjusting it with an app may reveal points that need more work, or give me a different feel to it).
There's no tried and true method to make art and palettes, but you do get more efficient at it when you apply the colors you've created in a context.
This being said, ckelsallpxls also raises great points about palette making, I agree with it as well.