Megan Rapinoe's brilliant strike put the hosts ahead after 33 minutes, but Houghton equalised three minutes later with a superb curling effort into the top corner after a well-worked free-kick routine.
Parris then gave England the lead in the second-half with a ruthless close-range finish that gave Adrianna Franch no chance.
The Lionesses were on the cusp of a famous result against the world’s top-ranked side but Tobin Heath’s goal meant Phil Neville and his players had to make do with a draw.
England have now taken four points form their opening two games and victory over Japan on Tuesday would see them win the competition.
That game will be shown live on BBC Four, with kick-off at 22:25 GMT.