Most PC's have space for 2 or more hard drives.
You could remove the hard drive from the older PC, and connect it into the new one.
The drive would occupy the next available drive letter, E,F,G....etc
You could then either drag and drop the photos to the larger hard drive, or burn to DVD.
On the back of the hard drive itself is a small jumper switch. You would need to set this to 'slave' (it is almost certainly currently set to 'master'
You can then reverse the process and put the HDD back into the original machine.