The short hearing was told the images related to stills rather videos but he had been caught out after police traced the payments he made for several of the images.
Barker had originally been arrested in 2003 but vanished a year later and was on the run until he walked into a London police station last month.
After his father, the much-loved star of Porridge, The Two Ronnies and Open All Hours, died in 2005 at the age of 76, detectives were waiting to question his son at the memorial service held at Westminster Abbey. However, Adam Barker did not turn up.
He also missed the funeral of his mother, Joy, who left an estate worth £2 million.
At the time of his disappearance he sent a typed note to his parents. Part of it read: "Dear Mummy and Daddy. Some time ago the police came to my house looking for images of under-age children on my computer. I was arrested.
"I must apologise for being foolish and thank you for being great parents. Rather than wait and find out what will happen, I will go away. You must understand that I won't be able to contact you for quite some time.
"Lots of love, Adam."
Barker pleaded to 20 counts of making indecent images of children at his home in Ealing, West London, between January 1999 and June 2003.
In charges one to 19, eight of the pictures are at level one, four at level two, three at level three and four at level four, a category which includes child rape.
Count 20 says he made 1,656 indecent images, namely levels one, two, three and four images of children.
Ravinder Johal, prosecuting, said: "The event is a serious offence and no doubt the probation will be making an assessment on the element of dangerousness."
He told the court the original indictment in this final count has been altered and reduced from 2,394 images to 1,656.
Judge Martin Edmunds QC said: "I shall order the indictment to be changed. Many of these are like offences."
Simon Ray, defending him, replied: "The figure was intended to reflect the fact that counts one to 19 reflect individual images. The total is 1,675 but count 20 is 1,656."
Judge Edmunds asked: "And these are still images as opposed to videos?" to which Mr Johal replied: "Yes, still images."
Judge Edmunds went on: "This is an old case going back to 2003."
Mr Ray replied: "That is how Mr Barker came to the attention of the authorities. It is right to say not all of the images were paid for but a proportion were and it is through these payment details he came to the attention of the police."
He went on: "Mr Barker has served the equivalent of a four month sentence of imprisonment, but my application is for a pre-sentence report."
Judge Edmunds said he would "not want to sentence a case of this nature without a report by an experienced officer."
He added: "I shall inform Mr Barker that he is subject to the Sex Offenders Act and there are particular conditions which will apply as and when he is at liberty. The period of that will depend on the sentence he receives."
Addressing Barker, he said: "This morning you have pleaded guilty to matters which normally carry custodial sentences but I will order a full pre-sentence report. Amongst other things it will look at issues of dangerousness. That will also look at the means by which the public can best be protected in the future."
Barker was remanded in custody and will be sentenced October 19.
He stayed away from his mother Joy's funeral when she died aged 78 after a short illness and he also missed his father's funeral in 2005 which was monitored by detectives.
