Parker, who will wear No. 50, will be available for Saturday evening's game against the D-backs.
The 27-year-old was 0-1 with seven saves and a 2.04 ERA in 16 relief appearances in Triple-A, holding opponents to a .133 average. He had 26 strikeouts in 17 2/3 innings and has allowed one earned run in his past 8 2/3 innings, spanning nine outings.
Parker made his Major League debut with the Cubs last season, allowing four earned runs in six innings (6.00 ERA) over seven appearances. He was selected by the Cubs in the 16th round of the 2006 First-Year Player Draft.
Dolis, 25, has thrown five scoreless appearances in three stints with the Cubs this season.
Burnett, 25, was claimed off waivers from Baltimore on Monday, made an appearance on Wednesday and was optioned to Iowa the following day.
Cash Kruth is a reporter for MLB.com. Follow him on Twitter at @cashkruth. This story was not subject to the approval of Major League Baseball or its clubs.