In the video, he stands outside his old school in South Yorkshire - the private Hill House School in Doncaster.
His parents were able to send him there after they started to sell stuffed Paddington bears.
He becomes emotional as he relives the memories he has of Yorkshire.
Clarkson, now 59, describes the first time he saw a Cortina 1600E in amber gold outside the gates of his old school.
To his surprise, the person sat in the car was his dad, Edward Clarkson
He said of the time: "It was beautiful and wondrous and exciting beyond words" and described how his knees actually buckled
As he recalled the memory, he described hairs standing up on the back of his neck just as they did 50 years.
He said he gave his dad a huge hug because he was so proud of him for having the E version of the Ford Cortina.
Edward died in 1994 and his mother, Shirley, in 2014.
Clarkson is known for masterminding the Top Gear format with the BBC before moving to The Grand Tour on Amazon's Prime Video streaming service, following a punch-up on the set of Top Gear.
Clarkson has appeared on the shows since 2001 alongside fellow presenters Richard Hammond and James May.
In the same episode, Clarkson was reduced to tears at the end of the show as the trio bid farewell to The Grand Tour's studio.
The show, Amazon's most popular programme, will return for a final set of globe-trotting special episodes, but the studio and audience format has been ended.
Clarkson later said the show was especially emotional because he never had the chance to say goodbye to Top Gear before his time on the show was cut short.