The lance bombardier, thought to be the most severely injured soldier to survive an attack in Afghanistan, went to the top of Wentworth Woodhouse in Rotherham.
The 35-year-old Bessacarr man, who lost both legs and suffered brain damage which affected his memory and speech in a bomb attack in 2006, said: "I'm very, very proud.
"Out of everyone they could have asked, they asked me."
Asked more generally about the state of his current health, Mr Parkinson said: "I'm really well."
He was speaking after sampling the stunning views from the top of Wentworth Woodhouse, looking out on huge statues and chimneys dating back to the Georgian times from on high.
The veteran was there to officially launch rooftop tours at the historic mansion.
Members of the public will soon be able to access the high roof by climbing 135 steps to the top, or by using the fully accessible lift that Mr Parkinson travelled in.
Wentworth Woodhouse was one of the great mansions of Georgian England but slid into neglect and disrepair during the second half of the 20th century.
The vast mansion is currently encased in 700 tonnes of scaffolding as contractors attempt to replace its six-tennis-courts-sized main roof as part of the first phase of a £130 million programme to save the building, whose 606ft frontage is wider than Buckingham Palace.
Last week, Downton Abbey creator Julian Fellowes revealed to Vanity Fair that a visit by King George V and Queen Mary to the mansion in 1912 inspired the storyline for his new film, a spin-off from the ITV series.
It was bought by the Wentworth Woodhouse Preservation Trust in 2017 for £7 million, prompting the start of the restoration project.
Dame Julie Kenny, the chairwoman of the trust, said the rooftop tours are designed to give visitors an insight into the property's surroundings, and to show where the investment is being spent.
She said: "Most brides don't want to have a wedding at the front of the house now that there's so much scaffolding, so we need to be flexible and fleet of foot to have different offerings for all sorts of people.
"This is an offering for those who like heights, who like to have a look around and for those who want to see something they'll only ever see once in their lifetime."
Explaining how the rooftop is being replaced to protect the inside of the property from rain damage, she said: "We inherited 60 buckets, we've now got about 10, so there's a significant reduction in the amount of buckets we use."
Dame Julie added that it was "really special" to have Mr Parkinson at the launch, saying: "It's great to show that we can actually cater to all disabilities here on the roof."