Let the distance between his house and park be 100 metres.
Lets assume he takes 30 mins daily. Hence his speed would be 100/30.
One day, if he wishes to save 1/3rd of 30 minutes, that is 10 minutes, he should cover the distance in 20 minutes in which case his speed would be 100/20.
Required increment in speed = 100/20 - 100/30
Percentage of increase in speed = (Required increment in speed / Original speed) X 100 % = ((100/20 - 100/30) / 100/30 )X100 % = 50%.