This is a continuation of notes on dynamics of quadratic polynomials. In this part we transfer the our prior geometric result to the parameter plane. To any parameter value c in the Mandelbrot set (which lies outside of the main cardioid and little Mandelbrot sets attached to it) we associate a "principal nest of parapuzzle pieces" and show that the moduli of the annuli grow at least linearly. The main motivation for this work was to prove the following:

Theorem B (joint with Martens and Nowicki): Lebesgue almost every real quadratic polynomial which is non-hyperbolic and at most finitely renormalizable has a finite absolutely continuous invariant measure.