  Close-in, sub-stellar companions to $\delta$ Scuti type stars present a
highly suitable testbed for examining how planetary-mass objects can influence
stellar pulsations. We aim to constrain the mass of HD 31221 b, probe its
atmosphere, and demonstrate how it affects the pulsational pattern of its host,
HD 31221. We made use of the available data from the short-cadence Transiting
Exoplanet Survey Satellite (TESS). We modeled the nine observed transits and
the out-of-phase variations, including Doppler beaming, ellipsoidal variations,
and the reflection effect. We also incorporated ground-based photometry from
the MuSCAT2 imager installed at the 1.52 m Telescopio Carlos Sanchez in the
Teide Observatory, Spain. We found HD 31221 b to have an orbital period of
$4.66631 \pm 0.00011$ days, with a radius of $1.32 \pm 0.14$ R$_J$ and a mass
of $11.5 \pm 10.3$ M$_J$ (from the ellipsoidal effect), making it consistent
with either a brown dwarf or a giant planet. As HD 31221 is a rapid rotator ($v
\sin I_\star = 175.31 \pm 1.74$ km s$^{-1}$), we deduced the spin-orbit
misalignment to be $\lambda = -121.6 \pm 14.4^\circ$ and $I_\star = 55.9 \pm
11.3^\circ$. The phase curve is dominated by the reflection effect, with a
geometric albedo of $1.58 \pm 0.50$. We also found evidence that HD 31221 is a
$\delta$ Scuti/$\gamma$ Doradus hybrid pulsator. There are three cases for
which the $3$rd, $85$th, and $221$st orbital harmonics almost exactly coincide
with peaks in the Fourier spectrum of the star, hinting at tidally perturbed
stellar oscillations. HD 31221 b is the third substellar object that is found
to be disrupting the pulsations of its host, following HAT-P-2 and WASP-33.
Additional photometric observations by CHEOPS and/or PLATO can be used to
further constrain its mass and provide a more in-depth analysis of its
atmosphere.
