On field Greenspace
open Field Calculator, set to Python Parser, and turn on Code Block.
In the Code Block:
def rework(leisure, greenspace):
if leisure == 'park' or leisure == 'golf_course':
return 1
else:
return greenspace
And in the expression:
rework(!leisure!, !Greenspace!)
The return greenspace
here leaves any existing values in Greenspace
. If this isn't here the other values in the column will be set to Null

You can also put it all on a single line (without using the Code Block) which is nice and tidy if this is the only possible combination, but if you have other values (which you want to return as 2
or 3
etc.) it will be easier to use the code block above.
1 if !leisure! == "golf_course" or !leisure! == "park" else !Greenspace!

In response to your comments, you can also try:
def rework(leisure, greenspace):
if leisure in ['pitch', 'golf_course', 'sports_centre']:
return 'sports'
elif leisure in ['grass', 'park', 'playground']:
return 'park'
elif leisure in ['garden', 'trees']
return 'vegetation'
else:
return greenspace