The discovery of superconductivity in infinite-layer nickelates has ignited stark interest within the scientific community, particularly regarding its likely unconventional origin. Conflicting magnetotransport measurements report either isotropic or anisotropic suppression of superconductivity in an external magnetic field, with distinct implications for the nature of superconducting order. In order to ensure a most suited model subject to subsequent many-body analysis, we develop a first-principles-guided minimal theory including Ni d_x2_−y2, La d_3z2_−r2, and La d_xy orbitals. Amended by the consideration of orbital-selective pairing formation, which emphasises the correlation state of the Ni 3d_x2_−y2 orbital, we calculate the superconducting ordering susceptibility mediated by spin fluctuations. We find a parametric competition between even-parity d-wave and, in contrast to previous studies, odd-parity p-wave pairing, which becomes favorable through a large quasiparticle weight renormalization for Ni 3d_x2_−y2 electrons. Our findings not only shed light on the distinctiveness of LaNiO₂ as compared to cuprate superconductors or nickelates of different rare-earth composition but also suggest similarities to other candidate odd-parity superconductors.