cook onions with sugar & salt, stirring often until onions brown...about 20 to 25 mins. Add a few tbsp. of water as the onions cook so as to not burn or stick to pan.
Fill a medium pot 3/4 full of cold water, add potatoes & parsnips. bring to a boil over high heat, reduce to a simmer & continue cooking potatoes & parsnips for 20 to 25 mins till they become fork tender, then drain.
Beat 1/4 of potato-parsnip mixture and 3 oz broth in a mixing bowl with an electric mixer on medium speed until smooth.
Repeat, alternating between potato-parsnip mixture and broth, until you've incorporated all.