Hi chaztait. I suspect that the new PC has serial ATA (SATA) connections to the motherboard, but the old DVD writer is the older parallel ATA. It's quite likely that the motherboard also has a parallel ATA header on it. Compare this click here with this click here, or for the newer round Parallel ATA cables click here
As Gongoozler has said, there should be an IDE port somewhere on the board and you can connect to that. IDE and PATA is different names for the same thing, look on your motherboard or in your manual for and IDE or PATA connector and connect to that.