Difficult to do that, as Parks was never in the military.
Parks didn't get the Congressional Gold Medal of Honor, Height did.
http://www.usmint.gov/pressroom/?flash=yes&action=press_release&ID=496 Dr. Dorothy Height Receives Congressional Gold Medal
President Bush presents honor to civil rights leader, grassroots organizer for lifetime work
WASHINGTON – President Bush today presented Dr. Dorothy I. Height, on the occasion of her 92nd birthday, the Congressional Gold Medal for her lasting legacy of public service and lifetime achievements in civil rights. The United States Capitol Rotunda was the site of the ceremony.
Prepared and struck by the United States Mint, the medal honors the work of Dr. Height, who has devoted her life to grassroots organizations created to fight racism, sexism, inequality, poverty and other societal ills. In the 1960s, she stood on the front line in the battles for civil rights for African Americans and equal rights for women.
But when Parks died, the President ordered the flags flown at half staff.
Apparently, if a President orders it, it can be done, like for the recent Hurricanes, Tsunami, VATech Shooting, Pope John Paul, Fort Hood Massacre, etc.....
http://en.wikipedia.org/wiki/Half-staff