There are two color palettes inside of PSP. There are the individual color swatches that are found on the Materials palette. And there are Image Color palettes. Image Color Palettes only exist in indexed images, images with 256 colors or fewer. These, when saved, are saved to the Palettes folder.
If your image is not already an indexed color image, you need to convert it to one. To do this go to Image -> Decrease Color Depth -> 256 color palette.
Once the image is indexed you can save the color palette by going to . Image -> Palette -> Save Palette. Give it a name. This will show up in the Palettes folder.
However, if you're talking about color swatches that appear on the Materials Palette, these are found in the Swatches folder.