We say that a triangle ${Delta}$ tiles the polygon ${ ho};if;{ ho}$ can be decomposed into finitely many non-overlapping triangles similar to ${Delta}$. Let ${ ho}$ be a parallelogram with angles ${delta};and;{pi}-{delta};(0<{delta}{leq}{pi}/2)$ and let ${Delta}$ be a triangle with angles ${alpha};{eta},;{gamma};({alpha}{leq}{eta}{leq}{gamma})$. We prove that if ${Delta}$ tiles ${ ho}$ then either ${delta}{in};({alpha},;{eta},;{gamma},;{pi}-{gamma},;{pi}-2{gamma});or;dimL_{ ho}=dimL_{{Delta}}$. We also prove that for every parallelogram P, and for every integer n $(where;n{geq}2,;n{ eq}3)$ there is a triangle ${Delta}$ so that n similar copies of ${Delta};tile;{ ho}$.