  The boundaries of the hyperbolic components of odd period of the multicorns
contain real-analytic arcs consisting of quasi-conformally conjugate parabolic
parameters. One of the main results of this paper asserts that the Hausdorff
dimension of the Julia sets is a real-analytic function of the parameter along
these parabolic arcs. This is achieved by constructing a complex
one-dimensional quasiconformal deformation space of the parabolic arcs which
are contained in the dynamically defined algebraic curves $\mathrm{Per}_n(1)$
of a suitably complexified family of polynomials. As another application of
this deformation step, we show that the dynamically natural parametrization of
the parabolic arcs has a non-vanishing derivative at all but (possibly)
finitely many points.
  We also look at the algebraic sets $\mathrm{Per}_n(1)$ in various families of
polynomials, the nature of their singularities, and the `dynamical' behavior of
these singular parameters.
