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Superconductivity and ferromagnetism are two antagonistic phenomena that combined can lead to a rich phenomenology of interactions, resulting in novel physical properties and unique functionalities. Here we propose an original hybrid system formed by a high-temperature superconducting film, patterned with antidots, and with ferromagnetic nano-rods grown inside them. This particular structure exhibits the synergic influence of superconductor (SC) - ferromagnetic (FM) stray fields, in both the superconducting behaviour of the film and the three-dimensional (3D) magnetic structure of nano-rods. We show that FM stray fields directly influence the critical current density of the superconducting film. Additional functionalities appear due to the interaction of SC stray fields, associated to supercurrent loops, with the non-trivial 3D remanent magnetic structure of FM nano-rods. This work unravels the importance of addressing quantitatively the effect of stray magnetic fields from both, the superconductor and the ferromagnet in hybrid magnetic nano-devices based on high temperature superconductors., FEDER, Severo Ochoa SEV2015-0496 program, Generalitat de Catalunya, Gobierno de Aragón, Grant from ICREA Academia, funded by the Generalitat de Catalunya, Ministerio de Economía y Competitividad (España), Unión Europea, Seventh Framework Program (FP7), European Cooperation in Science and Technology (COST), Depto. de Física de Materiales, Fac. de Ciencias Físicas, TRUE, pub