The zero curve is the curve that models actually use. Pricing a swap, a structured note or a bond portfolio means discounting each dated cash flow at its own spot-rate, and those come from the zero curve built by bootstrapping or read off treasury-strips.
It sits above the par-yield-curve when the curve slopes upward, because a zero's entire payment sits at the far end where rates are higher, with no early coupons pulling the average down.
Example: par curve 4.20% at 10 years, zero curve 4.31% at 10 years. Discounting a single $1,000,000 cash flow ten years out at 4.31% gives $656,000, not the $663,000 the par yield would suggest.
Related: spot-rate, bootstrapping, par-yield-curve, treasury-strips, forward-rate