I kept looking and Arthur BYRNE married twice
He married a second time on Oct 30 1888 to Elizabeth NOLAN
LINKThe record is not very clear but I think it gives Elizabeth's age as 26 implying she was born c1862 which would be approx 40 in 1901
The church record is not on line but states that Arthur BYRNE was married before and was from Ballinglen
Elizabeth BYRNE nee BYRNE died Aug 21 1885 aged 35 according to her death record
LINK #159
This implies she was born c1850
She and Arthur were the parents of
Joseph bap Oct 12 1871
Laurence bap May 1872
Margaret bap Aug 28 1874
Hugh bap Mar 06 1876
Maria bap Sep 06 1877
Patrick b Apr 25 1879
Catherine bap Oct 04 1880
Anne bap Mar 24 1882
Denis b Nov 01 1883
Arthur b May 31 1885
I still haven't located birth or baptism records for
Margaret or Elizabeth but without an estimated year of birth it is proving difficult with a common name - their mother may have been Elizabeth NOLAN if the births are after 1888. However, in light of your comment earlier
"Ditto for his sisters who had also would have left home by that stage (all in their 20's)" referring to leaving home before 1901 I would have expected their births to have been earlier that 1885 (aged 16 in 1901)
Based on both marriage records Arthur's father was Hugh
Based on the second record his mother was Margaret
This ties in with the baptism in Killaveny (just south of Ballinglen) on Mar 31 1846
of Art BYRNE to Hugh BYRNE and Margaret KEEGAN given earlier