Place chocolate chips in a small bowl. Heat milk until steaming, then pour over chocolate. Let sit for 2-3 minutes, then stir. If any chunks of chocolate remain, heat in the microwave for 30-second intervals until all the chocolate has melted. Mixture will be thin, like chocolate milk. Set aside and let cool to room temperature.
Using a food processor (or a gallon resealable bag + rolling pin), crush the Oreo cookies into crumbles, then set aside.
Using a mixer, beat Mascarpone cheese and powdered sugar on high until well blended. Reduce speed to low and gradually add chilled heavy cream and chilled Baileys, while continuing to beat mixture until mostly combined. Then increase speed to high and beat until stiff peaks form, about 5-7 minutes.
Divide cream mixture in half. The first half of the whipped cream will be left alone and is ready to use in the parfaits. The second half will be mixed with the chocolate. When ready, slowly fold in the chocolate mixture until completely mixed.
To assemble the parfaits: Using a pastry bag (or a Ziplock bag with the corner snipped off or just a spoon) add layers of the cream and crushed cookies. The parfaits pictured are layered as follows - chocolate mixture, then cream, then cookies, then cream again, then chocolate again, then cream again, and finally finish with any remaining cookie crumbles on top (see note below.)
Parfaits can be served immediately, but I highly recommend chilling them for 2-3 hours to firm them up. If chilling parfaits, add the cookies on top just before serving.