  Understanding electronic interactions in high-temperature superconductors is
an outstanding challenge. In the widely studied cuprate materials, experimental
evidence points to strong electron-phonon ($e$-ph) coupling and broad
photoemission spectra. Yet, the microscopic origin of this behavior is not
fully understood. Here we study $e$-ph interactions and polarons in a
prototypical parent (undoped) cuprate, La$_2$CuO$_4$ (LCO), by means of
first-principles calculations. Leveraging parameter-free Hubbard-corrected
density functional theory, we obtain a ground state with band gap and Cu
magnetic moment in nearly exact agreement with experiments. This enables a
quantitative characterization of $e$-ph interactions. Our calculations reveal
two classes of longitudinal optical (LO) phonons with strong $e$-ph coupling to
hole states. These modes consist of Cu-O plane bond-stretching and bond-bending
as well as vibrations of apical O atoms. The hole spectral functions, obtained
with a cumulant method that can capture strong $e$-ph coupling, exhibit broad
quasiparticle peaks with a small spectral weight ($Z\approx0.25$) and
pronounced LO-phonon sidebands characteristic of polaron effects. Our
calculations predict features observed in photoemission spectra, including a
40-meV peak in the $e$-ph coupling distribution function not explained by
existing models. These results show that the universal strong $e$-ph coupling
found experimentally in lanthanum cuprates is an intrinsic feature of the
parent compound, and elucidates its microscopic origin.
