Byrne, 22, has joined on loan from Championship club Nottingham Forest until the end of the season.
It is his second spell with the Bees this season, having scored four times in 19 appearances during a six-month stay earlier in the campaign.
He is joined at Underhill by 19-year-old Parsons, who has arrived on a month's loan from Crystal Palace.
Source: BBC Sport
Source: BBC Sport