Officers raided the home of Richard, 74, one of Britain's best-known entertainers, in August, with the search carried out in the glare of BBC cameras after the broadcaster was given advance warning.
Richard has denied any wrongdoing over the allegations, which date back to the 1980s and involve an underage boy. He has been interviewed under caution by police, but neither arrested nor charged.
The singer was on holiday when the raid was carried out and in a report, parliament's Home Affairs Committee criticised what it said was the unnecessary public nature of the police action.
"South Yorkshire Police's handling of this situation was utterly inept," said the committee's chairman Keith Vaz.
"Sir Cliff Richard has suffered enormous and irreparable damage to his reputation and he is owed an apology over the way matters were handled. We are not surprised that he wishes to sell his home."
South Yorkshire police said they had only allowed the BBC to film the raid because a BBC reporter was planning to break the story that Richard was being investigated. It said the broadcaster agreed to hold off in return for notification about the search of Richard's home in Berkshire, west of London.
The BBC said the police had volunteered information and that it would not have run the story if senior officers had contacted its executives.
"The force should have refused to cooperate and explained to senior BBC News executives why the premature broadcasting of a story, which they claimed the journalist threatened, would have prejudiced the investigation," Vaz said.
South Yorkshire Police said its actions were well-intentioned but ultimately flawed.
"We regret the additional anxiety which was caused to Sir Cliff Richard," it said in a statement.
Richard, born Harry Webb in 1940 and in his earlier career referred to as Britain's Elvis Presley, has had 14 No. 1 singles in Britain and is the only singer to have topped the UK singles chart in five consecutive decades, from the 50s to the 90s.
He shot to stardom in 1958 with "Move It" and followed up in 1959 with "Living Doll" while cementing his reputation as a heart-throb with starring roles in film musicals "The Young Ones" in 1961 and "Summer Holiday" in 1963.
With his backing group The Shadows, Richard was one of Britain's most successful performers in the pre-Beatles era of the late 1950s and early 1960s.
After the raid, he issued a statement saying he had been aware of rumours about him circulating on the Internet and that he would cooperate fully with police.
"The allegations are completely false," he said. "Up until now I have chosen not to dignify the false allegations with a response, as it would just give them more oxygen."
In a letter sent to the Committee during its investigation into the incident, Richard's lawyers strongly criticised the BBC.
It said: "We feel that it is necessary to record that, leaving aside the actions of the police, the actions of the BBC have also caused very serious harm to our client at a time when he had not been interviewed by the police, or of course arrested or charged."
