Parnell is the only major injury concern for the Proteas, who are likely to select Vernon Philander in his place as they open their preparations for next year's World Cup.
Captain AB de Villiers says he was not unduly worried by Parnell's absence going into the three-match series against the Black Caps and has backed his country's depth.
"Any of the 15 in the squad can find a spot in the main 11," De Villiers told reporters on Monday.
"I would even go so far as to say there are 20 to 25 guys in South Africa who can walk into this 11, that I know are good enough and that I have seen perform under pressure.
"I am very excited to see guys like Vernon Philander back, he has bowled well in the nets and bowled with a lot of skill, especially on wickets like these that we face in New Zealand."
Parnell is expected to be fit for selection for the second ODI at the same venue on Friday.
De Villiers also says he will look to be "creative" with his captaincy in the series, which will help to shape South Africa's thinking for the World Cup, which runs from Feb. 14 to Mar. 29 in Australia and New Zealand next year.
"This is the perfect opportunity for me and the team to be creative and adapt really quickly in the first 10 to 15 overs and hopefully find some weakness in the opposition," he said.
"It's the first international game at the Bay Oval and it's always difficult when there have been no international games in the past, but I am one of those captains that really enjoys thinking in the moment, it keeps me on my toes if I can be creative with my thinking."
South Africa have won nine of their 11 ODIs in 2014, which have included a series victory in Sri Lanka and success in a triangular tournament involving hosts Zimbabwe and Australia that was completed last month.
If the Proteas win all the three matches against New Zealand, they will reclaim the number one spot in the world rankings for the first time in five years.
(Reporting by Nick Said in Cape Town, editing by Nick Mulvenney)
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