We introduce a two parameter ($α, β>-1$) family of interacting particle systems with determinantal correlation kernels expressible in terms of Jacobi polynomials $\{ P^{(α, β)}_k \}_{k \geq 0}$. The family includes previously discovered Plancherel measures for the infinite-dimensional orthogonal and symplectic groups. The construction uses certain BC-type orthogonal polynomials which generalize the characters of these groups. The local asymptotics near the hard edge where one expects distinguishing behavior yields the multi-time $(α, β)$-dependent discrete Jacobi kernel and the multi-time $β$-dependent hard-edge Pearcey kernel. For nonnegative integer values of $β$, the hard-edge Pearcey kernel had previously appeared in the asymptotics of non-intersecting squared Bessel paths at the hard edge.