To calculate Key rate duration/Key rate DV01 for bonds, do we move the zero/spot curve or we move the par curve? Or either one is OK? Just want to know the industry standard.
You can do either. It depends on what you're trying to do and how you build your curve. If you're trying to match bond index duration, then shocking par curve is the way to go, because index providers, such as Barclays (now Bloomberg Barclays), Citi, and BofAML, all shock the par yield curve when reporting their option-adjusted duration statistics. However, a lot of "weird things" can happen when you do that (see "Duration Anomaly" and "Effective Duration versus Nominal Duration" by Bob Kopprasch); shocking zero curve tends to produce more intuitive results.
For hedging purposes, it may also be advisable to shock the forward curve, or individual input instruments you use to build the curve.