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How can I copy a palette from a photo in Photoshop? If Photoshop doesn't allow this, is there any application that does?

You could work with swatch libraries.

How to copy a color palette.
Note that in Photoshop, a color palette is called a 'color table'.
Step One: Open the image that has the colors you prefer for your palette (table).
Note that if you wish to give more weight to certain colors in this image, select the area(s) containing those colors before Step Two.
Step Two: Then IMAGE>MODE>INDEXED COLOR to convert the image to indexed color.
In the dialog box, choose ADAPTIVE and 256 OR FEWER COLORS. Click OK.
Step Three: Then IMAGE>MODE>COLOR TABLE.
In the dialog box, click SAVE, give the table a name, click OK, to save the palette (table) for later use.
Step Four: Then IMAGE>MODE>RGB to return to the RGB color space.

Then, to apply the palette to other pictures:
For each of the other images, open the image,
then IMAGE>MODE>INDEXED COLOR.
In the dialog box, choose CUSTOM and load the saved Color TAble. Click OK. They will be automatically adapted to the indexed color palette (table).

To simply access your saved color table, you can IMAGE>MODE>COLOR TABLE and load your saved palette (table). You can also use the saved palette (table) as a source for gradients or painting.
God bless you always, all ways,
Paul

Thanks, Paul. I'll try this immediately. What I'm trying to do is paint my photos using the colors of some of my favorite artists (painters).