Performances improved under Parker, who led them on a three-match winning streak in the league, but the former England international was unable to turn the team's fortunes around fast enough to ensure top-flight safety.
"Scott was brave to accept a very difficult challenge in February when appointed as our caretaker manager," Fulham owner Shahid Khan said in a statement.
"Since that time, day by day, whether in training or on matchday, we've undeniably become a better football club."
"Scott will now have another challenge to face, and that is to return Fulham to the Premier League. I fully believe Scott is more than up to the task."
Fulham spent close to 100 million pounds last summer ahead of their return to the Premier League but made a poor start under Slavisa Jokanovic, who was sacked and replaced by Ranieri after the club recorded one league win in 12 games.
Parker, who started the season as a first-team coach, said it was time to focus on an immediate return to the top-flight.
"I know how things work, the daily dynamics of team, and where improvements need to be made," Parker, who made 128 appearances for Fulham between 2013 and 2017, told reporters.
"Now, the challenge lies with me to move things forward. Ultimately, success comes from a culture and a desire - that needs to be our foundation. We'll drive a culture here that our fans will recognise.
Fulham host Newcastle United in their final game of the season on Sunday.
(Reporting by Hardik Vyas in Bengaluru; Editing by Hugh Lawson)
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