Why do I get a cold when I start my period?

Vagal reflex. It could be that when your period starts you experience an episode of vaso-vagal syncope which can result in a drop in pulse and blood pressure, as well as increased perspiration - all of which you could percieve as feeling "cold". Some women also have nausea which can also be caused by the vaso-vagal reflex associated with the menses.
Stay healthy. During menses, the immune system is depressed, meaning you're more likely to catch infections such as the common cold (caused by a virus). Keep your immune system healthy by getting plenty of rest all month long. Don't smoke, eat well, and get enough sleep.