In this paper, we propose two novel formulations for computing the subresultants of two Bernstein polynomials. Specifically, we derive subresultants under Bernstein basis in (i) determinant form and (ii) determinant polynomial form. To this end, we introduce two new subresultant matrices for Bernstein polynomials, whose determinant (for the first form) or determinant polynomial (for the second form) yields the subresultant of their standard monomial expansions. Notably, the resulting subresultant inherently retains the Bernstein form when expressed in polynomial form. The equivalence of these formulas to the standard monomial-basis one is rigorously established. Furthermore, we demonstrate an application of the proposed formulas in computing the GCD of parametric Bernstein polynomials. Unlike conventional methods, our approach avoids basis transformation, thereby eliminating numerical instability arising from such transformations.