It remains to be seen whether the concession makes the final broadcast - and it would be significant if it doesn't - but for Joshua to say that is an indication of how seriously he is taking the Parker challenge, and how well Parker conducted himself over a gruelling two days of media commitments after stepping off a plane from Las Vegas.
There were four film crews wanting interviews at the airport when he arrived in London for the fight announcement in January - and that was before the potential stress of a major press conference and long wait and face-to-face with Joshua at Sky TV's studios.
In the professional fight game, attitude and body language cannot be underestimated so the way Parker behaved suggests he won't be overwhelmed when walking out in front of a crowd of 80,000 at the Principality Stadium.
Parker's mental strength has always been seen as a key component of his success since he turned professional in 2012.
He is notoriously relaxed on fight day before making his way to the venue but he has a ruthless side, too, and both of these attributes have come to the fore in the build-up to the biggest fight of his life.
He will probably need a stoppage against Joshua to win because it's difficult to see the three judges scoring close rounds for Parker, the WBO heavyweight world champion, given he is fighting in enemy territory against a holder of two recognised world belts.
A recent example of how judges view things could be seen in the Wilder v Luis Ortiz fight last weekend for Wilder's WBC world title.
Wilder was up 85-84 on the judges' scorecards at the time of his stoppage victory in the 10th round against Ortiz despite the previously unbeaten Cuban appearing to win the first three rounds comfortably and taking a 10-8 round in the seventh when he poured on the pressure.
Most neutrals would have had Ortiz ahead on the scorecards by one or two rounds in that dramatic bout finished so comprehensively by Wilder after he was in serious trouble.
And while many boxing fans are turning their attention to a possible Joshua v Wilder unification fight later in the year, even Joshua's promoter Eddie Hearn is issuing a word of warning about the threat Parker poses.
"He sees Wilder as a very straightforward fight, so there's no fear," Hearn said in the United States after the American's victory.
"We know Wilder can punch, but I actually see the Parker fight as a tougher fight than the Wilder fight from a technical point of view.
"There might be areas of the fight where Parker outboxes and maybe out-speeds him and out-works him.
"He [Parker] is supposed to have this incredible chin, he's supposed to be incredibly durable. If he is, it's going to go deep and it's going to be a tough fight."
