Photos are rich data. There are ways to compress photos, but you may not want to do that.
First, compression breaks down into two categories: "Lossless" and "Lossy".
Lossy compression is accomplished by reducing the resolution of the photos. Essentially, the photo will never be as good after lossy compression. While that might not matter now, it probably will a few years down the line, when you'll want to blow the photo up for some purpose. The graininess will be greatly exasperated.
Lossless compression is also an option, but generally, lossless compression has little effect on photos. Many photo formats were designed to be relatively efficient, so there is not much advantage to be gained.
My recommendation is to go with the external hard drive idea.