Yesterday, Abraham and Channel 4 chairman Lord Burns were questioned by MPs on the Commons culture, media and sport select committee about Boyle's Tramadol Nights programme aired last year.
Committee chairman John Whittingdale noted that Channel 4 had "pretty vigorously" defended Boyle's actions on the show during an investigation by Ofcom into a complaint from Price.
The media regulator ruled in April that Boyle's comments were "highly offensive" and had a "straightforward focus" on Harvey and his disability. But Whittingdale asked Abraham why Channel 4 had still not issued a full apology.
Speaking at a hearing on Channel 4's 2010 annual report, Abraham told the MPs that it was "very difficult" to "deconstruct a decision around something as subjective as humour".
"He (Boyle) clearly works at the edge of taste but that is also the place where Channel 4 needs to be, but to be so in a responsible way," said Abraham.
"We obviously recognise that in that particular case a piece of humour that was contextualised in the programme late at night was then passed on in the media and out of context and did cause a reaction we had not intended."
Asked if he had seen, and approved, the joke prior to broadcast, Abraham said. "The referral process did come up to my level amongst many jokes in the series which were pushing the boundaries."
Channel 4 issued an open letter to Price explaining that the joke was made to "satirise the culture of celebrity" and was not intended to "focus the humour on a disabled child".
Abraham said that Channel 4 regretted the incident and would learn from the experience, but Conservative MP Louise Bagshawe said she found it "completely appalling" that the channel would not apologise.
Pressed on the issue, Lord Burns said: "I am content it was never intended to offend or cause distress to the son, that it was humour directed at the mother.
"But personally if it has caused distress to the son then obviously I am very sorry. It was not the intention and that was the key thing."
In a statement after the committee hearing, Katie Price said that Abraham continues to "add insult to injury" by refusing to apologise for Boyle's comments.
"How can this man be fit to run our Paralympic broadcaster? Even when parliament confront him on the issue he claims that the joke has been taken out of context. That is patently nonsense," she said.
"Why, when even his own chairman apologises, does he not take the opportunity too? His arrogance knows no bounds. Why won't he accept that jokes that are discriminatory of the disabled are in the same category as those that are discriminatory on racial grounds?"