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Amateur Peter, 46, beat reigning champion Neil Robertson to reach the third round, while Oliver, 21, defeated world number three Judd Trump.
They are the first father-and-son pair to make the last 32 of a ranking event since Geoff and Neal Foulds in 1986.
"We are on a high. If he's doing well, that's what I want," Peter said.
He told BBC Sport: "I will try as hard as I can and if I come out on top all well and good. If I don't, I will still be here on Tuesday with Oliver.
"If I do ok, it is just a bonus."
The Leeds-based duo have played together in the last 32 at PTC events but have never previously progressed to the same stage of a ranking event.
Peter plays Liam Highfield on Monday, while Oliver, who is 61st in the world rankings, plays Jimmy Robertson on Tuesday.
The pair are not due to meet until the semi-finals, "mainly because I have been going out early," said Peter, who added that he would not recommend putting any money on a Lines family meeting in the final four.
"I would say keep your money in your pocket," he said. "One of us might do, but I wouldn't like to say which one. There are a lot of big names in the top half of the draw. Ronnie O'Sullivan is in there and so is Mark Williams."
Peter, who fell off the main snooker tour this season, said watching Oliver is not normally stressful, but seeing his son struggle for form this season has been tough.
"It is probably the first time for the last two or three years that he has been low on confidence because he has risen up the rankings every year," Peter said. "Normally I just enjoy watching him play.
"But I have I always said to him that as long as he practises hard and has a first-class attitude, the results will come."
Oliver added: "It's weird. We are not used to winning. I will take playing him in the semi-finals.
"I have lost a few matches that I feel I should have won this season. I was struggling with my confidence before this tournament.
"But I have been practising very, very hard. Dad told me if I keep practising the way I am then my form will turn around. I am hoping it will be this week."
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Peter Lines says his shock run at the UK Championship is just a "bonus" and he would much rather see his son Oliver progress deeper into the tournament.