We analyze the pairing instability of an altermagnetic metal on a square lattice driven by an attractive nearest-neighbor interaction. This interaction enables multiple pairing channels, including even-parity extended $s$-wave and $d$-wave states, as well as two odd-parity $p$-wave channels. We verify that altermagnetic spin-splitting in the single-particle dispersion gives rise to finite-momentum pairing between electrons with unlike spins, in agreement with earlier predictions. Quite unexpectedly, this pairing typically emerges across multiple channels with mixed parity. Consequently, the resulting finite-momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting phase is expected to exhibit a multi-component order parameter featuring singlet-triplet mixing. We examine several forms of altermagnetism, specifically $d_{xy}$-wave and $d_{x^2-y^2}$-wave altermagnetic couplings, and present the corresponding phase diagrams. Additionally, we study triplet pairing between electrons with identical spins and find that it always occurs at zero center-of-mass momentum. Although it is unfavorable in the regime of weak altermagnetic coupling and low electron filling, it can dominate the phase diagram when the altermagnetic coupling is sufficiently strong. The influence of on-site attractive interactions on mixed-parity pairing is also explored.