the HID wiring should not touch anything to do with the parkers , or the wiring . Apart from taking one sides bulb input , there should be no issue. You have crossed or cut some wires somewhere or a loose connection is grounding somewhere ~

IMO take it all out and start again from stock. You should place the ballasts and the relay somewhere firm and hidden and the wires should be held close together and keep all connections clear from any possible debris etc