What does it typically mean when the plot of residuals vs. fitted values in a linear regression forms a parabola symmetric about the y-axis (for both convex and concave parabolas)? How can one infer from the shape of the residual plot the transformation that will need to be made (if one exists) to satisfy homoscedasticity?
If you're referring to a shape like this:
Then that doesn't indicate a problem with heteroskedasticity, but lack of fit (perhaps suggesting the need for a quadratic term in the model, for example).
If you see a shape like this:
that does indicate a problem with heteroskedasticity.
If your plot doesn't look like either, I think you're probably going to have to show us.