Parker was confirmed as West Ham's new head coach on Wednesday, November 4, following the departure of Nuno Espírito Santo, with the former Hammers midfielder returning to a club where he made 129 appearances as a player.
Jääskeläinen, who knows the demands of playing for West Ham, believes Parker's experience of the second tier makes him an ideal appointment, having previously secured promotion with Fulham, Bournemouth and Burnley.
"I think Scott Parker is the best choice, given what he's done already in the Championship," the Finn told Football Presse on behalf of kasinopartio.com. "He knows the football club very well, he knows the fans, he knows the demands of the football club. He's won promotion with three different teams already, and I think all the fans will be behind him.
"We know at West Ham you need to win in style, but I think the best connection you will create with the fans is going out, working hard as a team, fighting for everything and winning your football matches. That's the best way to connect the fans with the team."
Bowen can make the difference, but defending must improve
Parker inherits a squad containing captain Jarrod Bowen, whom Jääskeläinen identifies as the team's most important attacking threat.
The England international has consistently been West Ham's principal source of goals and creativity, but the former goalkeeper believes his contribution will become even more valuable if Parker can establish greater defensive solidity.
"He's the one who can win the matches for them. He's already shown it this season, and not just this season, all the time at West Ham, that he's the biggest threat. He can create chances, he can score goals.
"It will be easier for him if he knows that when he gets one or two goals in a game, his team can defend and keep more clean sheets, because it is crucial that the team functions well, especially in that league.
"It's important now that Scott Parker gets the defending working like he did with Burnley. Their goals-conceded record was unbelievable there."
Moyes departure was a mistake, insists former Hammer
Jääskeläinen also believes West Ham's recent difficulties can be traced back to decisions made higher up the club, particularly the departure of David Moyes.
The Scot delivered European success during his time in charge, but his exit was followed by a period of instability which has ultimately seen the Hammers drop into the Championship.
"Letting David Moyes go was a big mistake for West Ham. He did an unbelievable job there over the years and he's got all the experience. I know they were trying to look for maybe a different type of management, but it didn't really work for West Ham.
"I don't think the recruitment process has been the best since they won the European trophy, and I think it's just gone from bad to worse. It's important now that they go straight back up, because they've definitely got the squad."
He was also surprised that Nuno Espírito Santo had been unable to turn things around following his appointment, suggesting the Portuguese had struggled to maintain the support of his dressing room and the supporters.
"I am a little bit surprised that Nuno didn't get the team going, because he's got the experience from Wolves, and that is very important in the Championship, knowing how to win the league.
"I think he lost some of the players, and now you can see the fans didn't really back him anymore. They got a little bit frustrated that things didn't really work the way they wanted.
"When the fans turn against the team, it's very difficult to play in a full stadium when there's so much negative energy. I think that reflects on everybody [on the pitch], and I think the club made the right decision."
Sancho could be worth the gamble
There is also the possibility of further strengthening the squad, with Jadon Sancho among the names linked with a move to east London.
The former Borussia Dortmund, Manchester United and Chelsea winger has previously demonstrated the quality to make a significant impact, but Jääskeläinen believes West Ham would need to examine his circumstances carefully before committing to a deal.
"I think the club should look into it [signing Sancho]. You will look at his fitness levels, where they are at the moment, and you will look at what his motivation towards football is.
"Of course, if you get him anywhere near the level he was at when he was in Germany, then he will help the team a lot. But you need to look really closely at his fitness level, his motivation, and what he really wants from football."
For Parker, the immediate priority is restoring confidence, tightening up defensively and getting West Ham moving in the right direction.
Jääskeläinen believes the club has enough quality to achieve its objective. But his message is clear: hard work, collective commitment and a team the supporters can identify with must come before anything else.