He took possession of them before the start of the third day's play on Monday and said it was a "fantastic gesture."
His dad, David, a former England Test wicketkeeper, took his own life when his son was eight.
"I've had them for 39 years. My mum and dad took me to an Adelaide shopping centre and the English cricket team were there to meet," fan Andrew Johns told ABC radio Thursday.
"They had a little quiz -- they asked who the reserve wicketkeeper was for England and I shot my hand up, and said David Bairstow, and they gave them a pair of gloves and he signed them.
"I've had them sitting in a box for the last 39 years."
Johns said he made contact with Jonny Bairstow and arranged to meet him at the Adelaide Oval.
"I came in, I brought the gloves with me and sent him a message saying: 'I'm here'," Jones recounted.
"He came out two minutes later, and we had a good half an hour together which was wonderful. He was quite emotional to receive the gloves. It was really lovely."
Bairstow said the "fantastic gesture" meant a lot to him.
"It's something that is always very special. I've been fortunate enough to go all over the world, and all over the world people have some fond stories of dad," he told the broadcaster.
Bairstow has been in the headlines for the wrong reasons over a head-butting incident with Australia Test opener Cameron Bancroft in a Perth bar early on England's tour.
He insisted there was no malice intended and it had been blown out of proportion.
As a result of the distraction caused by the incident, the England team were placed on a midnight curfew for the rest of the tour.
First Published: December 5, 2017, 8:40 AM IST