Parkachik is well connected with daily bus service from Kargil. A bus leaves every day at 11 AM from Kargil town (Confirm timings in Kargil town) and reaches Parkachik around 3 PM. The same bus goes back to Kargil, the next morning around 7 AM.
But nothing beats having your own car though, especially if you plan to go ahead to Zanskar. Parkachik doesn’t have a fixed daily travel source to Zanskar. May be things are changed now. But there’s a good probability of finding a shared-taxi to Zanskar early in the morning.
Kargil is well connected with buses from both Leh and Srinagar.