The parallel port cables dabone refers to is the X1541 and variants. Google will help you find them. If all you want is to dump a few disks, and you have old PCs around to use Star Commander or something with, it works. If you want to spend a lot of time with your C128, you'll enjoy a flash device more.
Other alternatives, the C64 doesn't have a real serial port so that's out without extra hardware. The 1571 drive you have will read files from PC disks. You want the software Big Blue Reader, or Little Red Reader (found
here). If you get LRR up and running, you can use D64it/D128it (found
here to dump D64 images from disk. A D64 image ought to fit on an IBM formatted floppy, but you'll need another disk drive to write to as you read from the 1571. (Edit: actually I'm not sure this is true, it might be able read and write to the same drive, caching in RAM, I forget).
If you want to do serial, there's CBMLink(found
here). You do have to build a small serial adaptor though, so it's not the best option. I haven't done this so I can't tell you how it compares to an X1541.
I think that's all the options, besides buying a flash device (uIEC or 1541U).
Edit: No, there's one more. 1541EMU emulates an entire 1541 floppy with your PC. Apparently it's good enough to support fast loaders. Requires a PC with an parallel port though, and some soldering to build a cable. Doesn't look too bad, and could be a nice cheap substitute for a 1541-U if you want to play games with fast loaders.