  The second Painlev\'e equation with a large parameter ($P_{II}$) is analyzed
by using the exact WKB analysis. The purpose of this study is to investigate
the problem of the degeneration of $P$-Stokes geometry of ($P_{II}$), which
relates to a kind of Stokes phenomena for asymptotic (formal) solutions of
($P_{II}$). We call this Stokes phenomenon a "parametric Stokes phenomenon". We
formulate the connection formula for this Stokes phenomenon, and confirm it in
two ways: the first one is by computing the "Voros coefficient" of ($P_{II}$),
and the second one is by using the isomonodromic deformation theory. Our main
claim is that the connection formulas derived by these two completely different
methods coincide.
