  We show that the method in recent work of Roncal, Shrivastava, and Shuin can
be adapted to show that certain $L^p$-improving bounds in the interior of the
boundedness region for the bilinear spherical or triangle averaging operator
imply sparse bounds for the corresponding lacunary maximal operator, and that
$L^p$-improving bounds in the interior of the boundedness region for the
corresponding single-scale maximal operators imply sparse bounds for the
correpsonding full maximal operators. More generally we show that the framework
applies for bilinear convolutions with compactly supported finite Borel
measures that satisfy appropriate $L^p$-improving and continuity estimates.
This shows that the method used by Roncal, Shrivastava, and Shuin can be
adapted to obtain sparse bounds for a general class of bilinear operators that
are not of product type, for a certain range of $L^p$ exponents.
