We present explicit equations for the space of conics in the Fermat quintic threefold X, working within the space of plane sections of X with two singular marked points. This space of two-pointed singular plane sections has a birational morphism to the space of bitangent lines to the Fermat quintic threefold, which in its turn is birational to a 625-to-1 cover of P-4. We illustrate the use of the resulting equations in identifying special cases of one-dimensional families of conics in X.