Fresh from signing on loan with the club earlier in the week, both Burke and Evans played the full 90 minutes for Parkinson's men at Oakwell in City's goalless draw with Lee Johnson's Tykes.
The City chief said: "I thought it was a tough debut game to come into for Reece.
"Barnsley have a lot of good players in midfield and they like to play a possession-based game but they also have some strong players in aerial situations.
"(Alfie) Mawson, (Lewin) Nyatanga and (Marc) Roberts, plus the two strikers (Marley Watkins and Sam Winnall), are big threats from set-pieces - corners, long throw-ins etc.
"I knew it would be a tough test for Reece but he came through it with flying colours. He was calm in possession and looked very assured for us.
"With Evo (Lee Evans), I thought he got better as the game went on.
"The first half was all about us being disciplined and holding a good shape but as the game worn on in the second half, it opened up and that allowed Lee to show his quality on the ball.
"He gave us some real power in that midfield area and he looks like he could be a good acquisition for us."