Mr Burke said of his son's guilty plea: "It is what it is. It's the nature of the case".
He did not want to comment on what sentence he felt his son should receive. The maximum penalty for the charge is three years imprisonment or a $4000 fine.
The court heard yesterday that Burke, Mr Fyfe and another person were driving near Wanaka township, about 1am on December 19, when they spotted a deer at Mt Burke Station.
Burke and Mr Fyfe began pursuing the deer on foot in a manuka stand, both with rifles, said prosecutor Sergeant Ian Collin.
Mr Collin said Burke thought Mr Fyfe was behind him but Mr Fyfe had headed into the middle of the manuka, walking away from and slightly parallel to his friend.
"He stated he saw 'a small eye reflection, about 60m away ... that was moving away' ... [and] described it as a 'silver blue colour which was just like a deer eye reflection'."
After calling out to Mr Fyfe, Burke aimed through the scope on his .22 calibre rifle, fired one round at the reflection, and thought he had shot the deer.
He again shouted to Mr Fyfe and got no reply.
"The defendant went to the area ... and saw the deceased lying on the ground."
Mr Collin said Mr Fyfe showed no outward signs of life, and when Burke moved him into the recovery position he saw the bullet hole in the back of his head.
Mr Fyfe's father Grant Fyfe told the Herald it was too early to comment yesterday on the guilty plea.
Judge Kevin Phillips remanded Burke on bail until May 14 for sentencing, ordering a pre-sentence report with options for community and home detention sentences, although he warned it was no indication of the sentence.