The singer – who was in Portugal at the time of last week’s search – has denied an alleged sex crime against a boy at an evangelist event at a Sheffield stadium in 1985.
Police officers have been warned they could face disciplinary action over their dealings with the BBC.
“I think anybody not charged should not be named by the police, and shouldn't be reported in the newspapers either in my view,” said Parkinson on ITV News.
“I think the Cliff Richard case only highlights the feeling there is some kind of witch hunt going on.”
The veteran journalist criticised the BBC’s decision to cover the search without any charges or arrests having been made.
“I think the BBC did create an error in judgment, not in understanding the story and having the story and trying to follow it through, but in reacting to the story in a kind of way that would have done the red tops credit,” continued Parkinson.
"That's what wrong with the BBC, I think, on this one. It was the manner in which they chose to actually cover the event – if you can call it an event.
Cliff Richard: Life in pictures
Cliff Richard: Life in pictures
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Richard aged 18 at the Hulton Boys and Girls Exhibition at Olympia's Disc Theatre in August 1959 when he was lead singer of the Drifters
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Pictured in 1962 with members of the Shadows, his backing group. Next to him is Bruce Welch, who played rhythm guitar
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With his backing group, the Shadows, Richard became a massive pop star in the 1960s with a string of hit albums and singles including ‘I love You’ (1960), ‘the Young Ones (1961), Bachelor Boy (1962) and ‘Summer Holiday (1963)
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With his mother, Dorothy, and one of his sisters in 1964
Michael Stroud/Daily Express/Hulton Archive
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Outside the Royal Albert Hall where he represented the UK in the Eurovision Song Contest in October 1967
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Cliff Richard in 1968
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Cliff Richard poses in front of his disc awards in the 1970s
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On stage at the 1981 Royal Variety Performance, at the Theatre Royal, Drury Lane in London
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In an interview with the Daily Mail in 2008, Sir Cliff is reported as saying: "Of all the women I've known, there were only two that I've come close to marrying. One was Jackie Irving, a dancer whom I met in Blackpool. She was utterly beautiful and for a while we were inseparable. The other girl I came close to asking was Sue Barker, whom I met in 1982"
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Orchestrating a round of singing as rain delays play during the 1996 Wimbledon tennis championships
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Performing a duet with Flemish singer Helmut Lotti during a rehearsal of Lotti's album and TV broadcast 'Pop Classics' on 6 August 2003 in Brussels
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Performing on stage during the 'Intelligent Finance Cliff Richard Tennis Classic' at Birmingham National Indoor Arena on 18 December 2004
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Arriving with Brian May at the Royal Film Performance and World Premiere of 'The Chronicles Of Narnia' at the Royal Albert Hall on 7 December 2005 in London
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Unveiling a plaque of his handprints in Wembley Arena's Square of Fame on 9 November 2006 in London
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Sir Cliff Richard and The Shadows performs live on stage at the O2 Arena in London on 28 September 2009
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Cliff Richard attends a press conference to announce details of his new album at Gilgamesh on 7 March 2011 in London
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Queen Elizabeth II is introduced to Sir Cliff Richard backstage after the Diamond Jubilee, Buckingham Palace Concert on 4 June 2012, along with Sir Elton John, Sir Tom Jones, Paul McCartney, and Shirely Bassey
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Sir Cliff Richard at the 2012 US Open. The crooner claimed One Direction still compete with him to 'look cool'
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Performing on stage during the Diamond Jubilee concert at Buckingham Palace in 2012
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Cliff Richard attends the 'Willkommen bei Carmen Nebel' show at Volkswagen Halle on 23 November 2013 in Braunschweig, Germany
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Richard, 73, is one of the most successful British recording artists
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Sir Cliff, born Harry Webb, has sold an estimated 21.5 million singles in the UK and 250 million worldwide during a career spanning over 50 years
“I think there is a lot to be looked at and a lot to be learned from all that's been happening around that particular kind of area.”
Parkinson noted that in the case of Rolf Harris (who was imprisoned for his sexual abuse crimes) the media arrived at his Bray home before the police.
“It's not right. Particularly at that point, he was not charged with anything,” he said.
"I just feel that they should tread more softly and we should be more considerate of everybody's feelings and claims and rights in this.
“We should pursue people, of course, who have done wrong. That is indisputably the police's job.”
Richard’s family also defended the singer this weekend.
“I know my uncle is innocent,” said his niece, Linzi Jolin, while his cousin, Garth Gregory, described the raid as a “witch hunt”.
His fans have also pledged their allegiance, with a Facebook campaign having been launched to get his 1992 track, “I Still Believe In You”, to number one in the charts.