I don't have a parts bike, but I do have a C-100(50cc), C-102(50cc electric start), and CM-91(90cc), three of the original four Cubs, the only one I am missing is the C-65(70cc).
In addition to the bikes I also have some parts, NOS motor parts, and lots of used parts. I am down in Springfield, but a lot of the parts are in Champaign right now.
Check and see what model you have, on the riders left side of the bike is the VIN, just in front of where the sidecover should be, the first numbers will be C100 or C102 if it is a 50cc, another way to tell the difference is the electric starter on top of the motor on the 102, how do you know it is a 1959? there were no years put on them, so it is really a guessing game to the year, but it can be narrowed down by the VIN, also there were the later models of the CA-100 and CA-102 that were near identicle.