"That we recognize as brethren, and welcome to our fellowship, all who have been immersed (by whomsoever) after their acceptance of the same doctrines and precepts."
Why were those words in parenthesis put there?
First, they recognise what is sometimes called "the priesthood of all believers". In other words, baptisms don't have to performed by ordained clergy in order to be valid. Anyone can perform a baptism.
Second, they put the emphasis on the person being baptised, not on the baptiser. In fact, the person performing the baptism doesn't necessarily have to have the same faith or beliefs as the person being baptised. For quite some time the identity of the person who re-baptised John Thomas was unknown (although some recent research has brought to light some convincing evidence that it was John Tomline Walsh, who never became a Christadelphian*).
Third, these parenthetical words allow for the possibility that some people may have been baptised while members of other denominations while believing "the same doctrines and precepts" as Christadelphians.
From my own family history there is an interesting account of three people who were accepted as Christadelphians although they were baptised in another church.
The Christadelphian Shield for June 5, 1941 has this intelligence from Granville (NSW) ecclesia about the baptism of my late father:
This was followed by a record that Granville ecclesia had received into fellowship three other people (including my late grandmother and late uncle) who had "previously been baptised into Christ believing these principles and doctrines". My grandmother had, in fact, been baptised in the Church of Christ and my uncle in an independant Christian Assembly, but after joining the Christadelphians they were members of Granville and then Yagoona ecclesias until they died, without ever being re-baptised by Christadelphians. The report says these three "had not met in fellowship with any ecclesia" although they had, in fact, been meeting and were "in fellowship" first with the Church of Christ and then with an independant Christian Assembly for some time before joining the Christadelphians.
* Peter Hemingray John Thomas - His Friends and His Faith 2003 p.145