Tommy
Thankyou for this prompt reply. We think the brother left Sweden permanently too. I always imagined he stayed in the US but I have no proof. Another anecdote in the family is that Charles and his brother were on board a ship in New York when Jenny Lind visited and he commented how busy the harbour was.
Another relation has also been searching. Some of this i got from her.

Sten Aminoff made a thorough investigation of the NZ Swedes in his book "Svenskarna i Nya Zeeland" (The Swedes in New Zealand). Charles Parker is there with the following information:

Parker, Charles Francis
Born 1834 in Sweden (note date, correction of my first post Charles born 1834 or 1836 and his brother .born ?1838)
Arrived in NZ in 1856
Naturalized Wellington 3 July 1863
Died Wellington 15 Dec 1917
Farmer
Married Wellington 9 April 1860 to Harriet Ellen Philips, 17, spinster

Apparently There is another book that has almost all Swedes who emigrated to the US 1820-1850: "Swedish Passenger Arrivals in the US 1820-1850" by Nils William Olson and Erik Wikén. There is no Parker in that book.

There is also a Charles Parker, with home parish listed as England, who mustered out of the Gamla Karleby sjömanshus on 5 August 1841, i found on Arkivdigital when i had a Short term subscription. Could it be the father of ggfather Charles - but there isnt much information there.

Thankyou again very much for your interest and your time spent looking for information. I'm still hopeful something will one day turn up.