I seriously hope the OP made a typo when he said he wanted to parachute on the lunar surface. There is no air on the Moon/Mun. NASA even has to be careful on Mars missions because the air is not enough to slow the craft down.
No. That is impossible. Mammals descended from reptiles and even lived alongside the dinosaurs. Many dinosaurs had to protect their eggs from small, possum like mammals. Some of these mammals began to go up into trees after the KT extinction and later evoled into primates, which then evolved into hominids including us.