In this paper, the author explores the sources of European Union Law that regulate one segment of parental responsibility - the right of access to a child. The focal point of research is the transition from the conventional (interstate) regulation of judicial cooperation in marital disputes and parental responsibility issues to the regulation enacted by the European Union institutions, with specific reference to the Brussels II bis Regulation. First, the author briefly points out to its relationship with other relevant international law sources regulating this subject matter: the Hague Convention on the Civil Aspects of International Child Abduction; the Hague Convention on Jurisdiction, Applicable Law, Recognition, Enforcement and Cooperation in the Field of Parental Responsibility and Measures for the Protection of Children; and other international sources of law. Then, the author examines in more detail its relationship with the Brussels II bis recast Regulation, which will be applicable as of 1 August 2022. In addition, the paper includes an analysis of the first case in which the Court of Justice of the European Union (CJEU) decided on the application of the Brussels II bis Regulation, at the request of granparents to exercise the right of access to the child. On the issue of determining the competent court which has jurisdiction to decide on how this right shall be exercised, the CJEU had to decide whether the competent court is determined on the basis of the Brussels II bis Regulation or on the basis of national Private International Law rules. This paper is useful for the professional and scientific community because it deals (inter alia) with the issue of justification of adopting a special source of law at the EU level, which would regulate the issue of mutual enforcement of court decisions on the right of access to the child. This legal solution was proposed by the Republic of France, primarily guided by the fundamental right of the child to have contact with both parents.