Let \(f : \widehat{\mathbb C} \to \widehat{\mathbb C}\) be a rational map of degree \(d \geq 2\), and denote by \(f^n\) the \(n\)th iterate of \(f\). The Fatou set \(F(f)\) is the maximal open set in which the sequence \((f^n)\) of iterates is a normal family in the sense of Montel, while the complement of \(\mathcal{F}(f)\) in \(\widehat{\mathbb C}\) is the Julia set \(J(f)\). A zero of \(f'\) or a multiple pole of \(f\) is called a critical point of \(f\). A point \(z_0 \in \widehat{\mathbb C}\) is called an eventually periodic point of \(f\) if there are integers \(m \geq 0\) and \(p \geq 1\) such that \(f^{m+p}(z_0)=f^m(z_0)\). If all critical points of \(f\) contained in the Julia set \(J(f)\) are eventually periodic, then \(f\) is called geometrically finite. The Fatou set of such a rational map cannot contain Siegel disks or Herman rings. A perturbation \(f_\varepsilon \to f\) of \(f\) is a family of rational maps \(f_\varepsilon\) of degree \(d\) with \(\varepsilon \in [0,1]\), \(f_0=f\) and \(f_\varepsilon \to f\) as \(\varepsilon \to 0\) uniformly on \(\widehat{\mathbb C}\). If \(f_\varepsilon \to f\) is a perturbation of \(f\), and if for \(\varepsilon \in [0,1]\) there is a homeomorphism \(h_\varepsilon : J(f_\varepsilon) \to J(f)\) with \(h_\varepsilon \circ f_\varepsilon = f \circ h_\varepsilon\) on \(J(f_\varepsilon)\), then \(h_\varepsilon\) is called a topological conjugacy between \(f_\varepsilon\) and \(f\) on their respective Julia sets. If \(h_\varepsilon\) is not a homeomorphism but merely continuous and surjective, then \(h_\varepsilon\) is called a semiconjugacy. The author considers so-called horocyclic perturbations of a geometrically finite rational map \(f\) into another geometrically finite rational map \(f_\varepsilon\) which preserves the critical orbit relations with respect to the Julia set of \(f\). He constructs a semiconjugacy or a topological conjugacy between the dynamics of \(f\) and \(f_\varepsilon\) on their Julia sets. For hyperbolic rational maps, the existence of such conjugacies is well known from results of \textit{R.~Mañé}, \textit{P.~Sad} and \textit{D.~Sullivan} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 193--217 (1983; Zbl 0524.58025)]. But this theory is not applicable here, since geometrically finite rational maps may have parabolic cycles. The results in this paper give a partial answer to a conjecture of \textit{L. R.~Goldberg} and \textit{J.~Milnor} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, 51--98 (1993; Zbl 0771.30028)].