How can I plot a parametric equation with ±?

This formulation might be useful to you.

±b_ := {b, -b}

ParametricPlot[Thread[{x, ±x^2}], {x, 0, 1}, Mesh -> 30, 
 PlotPoints -> 50, AxesOrigin -> {0, 0}]

Notes

• This will expand appearances of ±value outside of plotting as well. If you need ± to persist in data this expansion should be restricted. We can write a custom plotting function, say parametricPlot, that performs this and other expansions, then hands off the data to ParametricPlot.

• The form used above results in shared styling between the positive and negative branches. Reference: Plot draws list of curves in same color when not using Evaluate

You can just do

ParametricPlot[{{x, x^2}, {x, -x^2}}, {x, 0, 1}, Mesh -> 30, 
 PlotPoints -> 50, AxesOrigin -> {0, 0}]