  The lifted horn map $h$ of a holomorphic function $f$ with a simple parabolic
point at $z_0$ is well known to be a local conjugacy complete invariant at
$z_0$; this is a classical result proved independently by \'Ecalle, Voronin,
Martinet and Ramis. Landford and Yampolski have shown that, if two functions
$f_1, f_2$ with simple parabolic points at $z_1, z_2$ are globally conjugate
over their immediate parabolic basins, with the conjugacy and its inverse
continuous at $z_1$, resp. $z_2$, their horn maps must be equal as analytic
ramified coverings. In this article, we introduce a notion of local conjugacy
over immediate parabolic basins, and show that two functions $f_1, f_2$ with
simple parabolic points are locally conjugate over parabolic basins (without
the hypothesis of continuity) if and only if the non-lifted horn maps
$\mathbf{h}_i$ are equivalent as ramified coverings. This result is a first
step to understand better invariant classes by parabolic renormalization.
