Artin groups are given by a finite set of generators $S$ with the presentation: $$A = \langle S \mid \underbrace{s_i s_j s_i \dots}_{m_{ij} \text{ letters}} = \underbrace{s_j s_i s_j \dots}_{m_{ij} \text{ letters}},\; \forall\, i \neq j \rangle.$$ These relations generalize those that define Coxeter groups, but without requiring the generators to have finite order. A parabolic subgroup of an Artin group $ A $ is obtained by considering a subset $T \subseteq S $ of generators and taking the subgroup generated by $T$. These subgroups play a fundamental role in the study of the topological and algebraic properties of Artin groups. In 1997, Luis Paris, building on the work of Kramer for Coxeter groups, proposed an algorithm that efficiently determines whether two parabolic subgroups are conjugate in the Artin group. Dyer groups, on the other hand, form a family that generalizes both Coxeter groups and RAAGs (Right-Angled Artin Groups). They admit a uniform solution to the word problem in both cases (Coxeter and RAAG) and allow the definition of parabolic subgroups in a manner analogous to that of Artin groups. In this talk, we will present an algorithm, based on the works of Paris and Kramer, that decides whether two parabolic subgroups of a Dyer group are conjugate. This is a joint work with Marina Salamero (Universidad de Sevilla) and Mireille Soergel (TU Berlin).