  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 $\PP^4.$ We
illustrate the use of the resulting equations in identifying special cases of
one-dimensional families of conics in $X.$
