In $\triangle ABC$, $AB=AC=35$ and $BC=24$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?

Since $DE\parallel AC$, $\angle DEB=\angle C=\angle B$, so $DB=DE$ and $AD+DE=AD+DB=AB$. The perimeter of $ADEF$ is therefore $2AB$.

Perimeter $=2(35)=\boxed{70}$.