Rabada, on his first global visit to Lord's, had figures of 4/39 while Parnell took 3/43, the other three wickets falling to Keshav Maharaj.
"This result ensures we go into the Champions Trophy without big heads", Bayliss told BBC's Test Match Special.
Earlier, South Africa won the toss and chose to field first.
A collapse of four wickets in seven balls, spearheaded by Rabada's pace and clever seam bowling from left-armer Parnell, meant England made their worst start to a one-day innings at Lord's.
"I don't think it was an ODI wicket".
Watch highlights of the third ODI between England and South Africa on Sky Sports 2 tonight from 8pm. "They did not give us anything to hit and if they did we managed to nick it. Credit goes to South Africa, they came back really well".
Three days out from their competition opener against Bangladesh, England were given a rude awakening after slumping to 20 for six after five overs in the third one-day worldwide against South Africa at Lord's.
Instead, captain Morgan - who plays domestic cricket for Middlesex at this ground - questioned the quality of the wicket prepared for this game.
"I'd be disappointed if we did, because any team (would have) the potential to win or lose a game on a toss in a major tournament - which is hard to take and makes it one-sided, which I don't think is good for anybody".
"We had a lot of time to talk about it as a batting unit watching our bowlers bat which should never be the case", said Morgan.
Morgan has previously stated that, despite his impressive form, Bairstow is unlikely to feature in England's strongest XI for the Champions Trophy. 'He's putting on a lot of pressure and that's what we want from him.
"It's the hardest thing telling Jonny he's not playing when he's done nothing wrong and he scores a huge amount of runs", Morgan said.
'As regards selection, Jason is the No.1 pick at the moment.
Morgan also eased fears about England's various injury concerns, saying Ben Stokes, Chris Woakes and Moeen Ali would all be fit for Thursday after missing this match. He and Alex Hales have been our opening pair for quite a long time. "They have had ups and downs but ultimately they have played in the fashion that we have played as a team and they have been very important to that".
It was just the kind of performance England didn't want ahead of their ICC Champions Trophy opener, and just the kind South Africa were looking for. Four out of the top five England batters were out fishing at deliveries they should have shouldered arms to.
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