This one may depend on who you believe, but bookmakers have Wrexham as the slight favourites to win the tie.
The Red Dragons have home advantage and now find themselves in a higher division than the Bluebirds.
Parkinson said: “Are we favourites? Home advantage, a division above them, I imagine the bookies will have us as slight favourites, but I’m not concerned about that either way.
“All we’re concentrating on is our performance level and picking the team and bench we feel can give our fans the bragging rights in this Welsh derby.”
Barry-Murphy was similarly diplomatic when it came to giving a verdict on which side are most fancied to emerge victorious.
“Well presumably they’re (Wrexham) doing things in their own way. It’s not for me, it’s none of my business what they do,” said Barry-Murphy.
“But what I’ve seen from them clearly already is they have an experienced team chock full of players who’ve been there, done it at that level, if not higher, and they’ve invested heavily in that.
“But our squad we believe is very capable of competing with teams who are at Championship level. So now we get a chance to go on Tuesday and show them, and that’s where our intention will be.”