In the final match to be held in Tokyo this season -- with the last two Sunwolves home games in Hong Kong and Singapore -- flyhalf Parker amassed 36 points and winger Saumaki scored a hat-trick of tries to inspire a much-needed victory.
Tries from Parker and lock Grant Hattingh, as well as five penalties from the former, gave the Sunwolves a 15-point lead at halftime and Jamie Joseph's men continued to punish the 2011 Super Rugby champions after the break.
Three second half scores from Saumaki and a penalty try sealed the comfortable victory.
"For me, as the coach of this team, this result wasn't a surprise," stressed Joseph.
"Every week we had been improving but we hadn't been able to get over the line and get a win so today, for me, the team put everything together in the same match."
Joseph made five changes to the side that lost to the Wellington Hurricanes two weeks ago, including a return for talismanic Brave Blossoms skipper Michael Leitch from injury.
However, it took some time for the revamped lineup to make an impact as converted tries from Brandon Paenga-Amosa and Ben Lucas in the first 22 minutes gave the Reds a 14-9 lead.
Parker slotted home four penalties in the first 28 minutes to keep the Sunwolves in the contest before the match turned on the half hour mark when Hattingh went over for the home side's first try of the day.
Referee Egon Seconds awarded the score after almost five minutes deliberation as the television match official checked for a possible forward pass in the build-up.
After that it was all Sunwolves, with Parker touching down three minutes later following some slick handling from centre Michael Little.
The Reds were punished for further indiscretions at the breakdown as Parker added two more penalties either side of halftime before Saumaki went over in the corner to give the hosts a 39-14 lead with more than 30 minutes to play.
As the Sunwolves defence tired, Reds number eight Angus Scott-Young pulled a try back for the Australian team on 66 minutes but just moments later, Duncan Paia'aua's high tackle resulted in a yellow card and a penalty try for the hosts.
In the final 10 minutes, Saumaki scored two more tries to complete an emphatic victory.
This was Joseph's penultimate game in charge of the Sunwolves before he turns his attention to the Japan national team and their June tests against Italy and Georgia.
Reds head coach Brad Thorn said they had got "exactly what we deserved" from the game.
"It is a long way to come to perform like that. To dish that sort of performance up is pretty damn frustrating," he added.