  In this article, we study analyticity properties of (directed) areas of
epsilon-neighborhoods of orbits of parabolic germs. The article is motivated by
the question of analytic classification using epsilon-neighborhoods of orbits
in the simplest formal class. We show that the coefficient in front of
epsilon^2 term in the asymptotic expansion in epsilon, which we call the
principal part of the area, is a sectorially analytic function of initial point
of the orbit. It satisfies a cohomological equation similar to the standard
trivialization equation for parabolic diffeomorphisms. We give necessary and
sufficient conditions on a diffeomorphism f for the existence of globally
analytic solution of this equation. Furthermore, we introduce new
classification type for diffeomorphisms implied by this new equation and
investigate the relative position of its classes with respect to the analytic
classes.
