Nottinghamshire bowler Stuart Broad moved on to 384 international dismissals – one ahead of Botham - on Saturday as England thrashed West Indies inside three days in the first Test at Edgbaston.
He now has only 35-year-old team-mate James Anderson (492) in front of him.
Speaking to BBC Five Live's Sportsweek programme, former England batsman Chris Broad said: "(I'm) immensely proud, obviously. He's been a shining light in the family for his career since he first started.
"We're all very proud of his achievement and to get to a person like Ian Botham and his record is a monumental achievement. When he's bowling, he just looks a dream."
The younger Broad made his Test debut against Sri Lanka in December 2007 and has since played a further 106 times.
Father Chris, who represented his country between 1984 and 1989 and played alongside Botham, believes the 31-year-old will continue giving batsmen headaches for the foreseeable future.
"You look at Jimmy Anderson, you look at Stuart, they are as fit now as they were when they were 21-year-olds," he added.
"Fitness is not going to be an issue for either of these bowlers really in ending their careers - it'll be about form, it'll be about losing that little bit of extra zip and not taking wickets.
"As long as both of those bowlers – Jimmy I thought looked fantastic (on Saturday), and in fact throughout the game - continue to hold on to that nip, that little bit of zip off the pitch, and stay fit, then they can go on for as long as they want.
"I just wonder whether, like great partnerships, when one goes, maybe the other one will go."
Chris, 59, also revealed that his son had a few drinks with Botham after surpassing the 61-year-old's wicket total.
"I heard (on Saturday) that he and Ian Botham were going to share a quiet drink – well, I know what Ian Botham's quiet drinks are like...But hopefully it didn't end up like that (on Saturday night)."
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