Newton's laws are invariant under time reversal transformation $$ t \longrightarrow -t $$ for time-independent potentials.
But Hamilton-Jacobi equation is too an equivalent description of classical mechanics which is not invariant under this transformation because of being 1st order in time.
Where is the fallacy?
The Hamilton -Jacobi equation:
$$\frac{\partial S}{\partial t}=-H$$
with:
$$H=\frac{\partial \mathscr L}{\partial \dot{q}}\,\frac{d q}{dt}- \mathscr L$$
you get:
$$\frac{\partial S}{\partial t}=-\frac{\partial \mathscr L}{\partial \dot{q}}\,\frac{d q}{dt}+ \mathscr L\tag 1$$
reverse time $t\mapsto -t$ we obtain for eq. (1)
$$-\frac{\partial S}{\partial t}=+\frac{\partial \mathscr L(-t)}{\partial \dot{q}}\,\frac{d q}{dt}+ \mathscr L (-t)\tag 2$$
so equation (2) is equal equation (1), if $\mathscr L (-t)=\mathscr L (+t)$ this is the case in your for your question.
you get the same results for Euler- Lagrange equations.
Assuming the Hamiltonian is invariant under $T$-transformation, the Hamilton-Jacobi (HJ) equation is also invariant, since Hamilton's principal and characteristic functions change sign.