"What a band!" roars the English singer-songwriter when talk turns to The Rumour, his backing group in the late 70s and, since an unlikely reunion in 2011, his cohorts once again. "There's really not many bands this good."
We shall soon find out. Following the release of an excellent album of new material, Three Chords Good, and a successful string of gigs in the US, later this month the six-piece will finally play their first shows in Britain in more than 30 years.
Parker decided to reunite The Rumour two years ago on something of a whim, sending a flurry of late night emails which received a positive response from his five former band-mates.
"For some reason it seemed the right thing to do," he says. "It was an unexpected thing and now it's this ball that keeps rolling. I'm not sure I have full control over anything any more, but it's exciting.
"The shows in the US have been fantastic but we haven't played in the UK since our last tour, so it's all a bit shocking for us." He laughs. "But then it was shocking from the beginning."
Parker and The Rumour found their footing in the London pub rock scene of the mid-Seventies, the fertile pre-punk breeding ground which also cultivated the early careers of Elvis Costello, Ian Dury, Joe Strummer and Nick Lowe. It turns out he has no great fondness for the terminology.
"Pub rock? It was a shock to see this thing growing around me like a cancer," he says. "I thought I was just doing classical styles that I'd dredged back up to the surface: pop, soul and Dylan thrown together. I was astonished to find it being lumped into pub rock." Because The Rumour included Martin Belmont, Bob Andrews and Brinsley Schwarz, past members of pub rock stalwarts Ducks Deluxe and Brinsley Schwarz, Parker says he found himself "guilty by association. I honestly thought Brinsley Schwarz were a German heavy metal band, I knew nothing about it. I think the term did play against me, because Americans had no idea what pub rock was. It sounded to them like a guy sitting at a piano singing about boiled beef and carrots: Olde English music hall. We were just a rock and roll band."
Indeed they were. In their late 70s pomp The Rumour were renowned for the pace and sweat-soaked intensity of their live shows. Parker's taut snarl pre-dated Costello, while the band possessed a soulful, full-tilt intensity which recalled Bruce Springsteen and Van Morrison. "It was like being on a rocket in the 70s," Parker says. "Before we did any gigs this time around we listened back to some old recordings and we went, 'Well, we're not going to do that, are we?'
"Our claim to fame was that we were Britain's fastest band. I can't speak for everyone but it wasn't chemically induced, it was just how we were. My lead was intensity and the band followed it perfectly. Then punk rock hit, and without consciously saying it we started playing even faster."
Those days are gone, and Parker doesn't necessarily miss them. "In the early days we used to swing, but we lost it," he says. "By the time of [1979 album] Squeezing out Sparks it had become a kind of hyperventilated rock.
"Now we want to bring out the great musicianship of the band, because the songs have been missed a lot of the time. It's party my fault. We cranked it up on stage, and people still think of us as this band who would kick their teeth in."
Although Three Chords Good is a more laid back affair than previous Rumour albums, it is recognisably the work of the same band that recorded their debut, Howlin' Wind, back in 1976. "The main influence is still soulful roots music," says Parker. "It was there on my first album and it's there on the new one. I can move in different directions from that, but that's always the template."
Lyrically, he is still drawn to the exposed nerve. Having been resident in the States on and off since the early 80s - "I've got my foot in both doors, and always have" - new songs like Coathanger, Arlington's Busy and Snake Oil Capital of the World confront American right-wing fundamentalist views on issues such as race, US foreign policy and abortion. "I don't want to write them, it just happens," he sighs. "I'm like Homer Simpson: 'Doh! Why have I done this?'
"I always regret it. In America there's a lot of people who still can't believe that there's a black man in the White House, it's like the 60s have come home to roost.
"The right-wing bloggers have gone to town on Coathangers, which is about abortion, and I don't really want that. I want a quiet life, but I have to stick up for my songs."
Parker has just returned from what he calls his day job - playing solo shows all over Europe, America and Japan.
The days when he was given £300,000 to make an album and it took "three days to get a snare sound" are long over.
Nowadays he has to diversify to survive. In the past decade there has been a book of short stories -Carp Fishing on Valium, recently updated with three new tales - and a novel, The Other Life of Brian. Last year he even made a brief acting appearance in Judd Apatow's comedy This is 40.
Despite the excitement and attention focused around The Rumour's reformation, playing live, and alone, remains his bread and butter. "I've exploited the solo aspect," he says. "I've become good at it. It's like a bit of stand up comedy with a bit of music, and it's a very creative way to approach your back catalogue.
"That's the way to make a living. I didn't think I'd be 62 and people would still be telling me I have to go out and do gigs, but there are all these other angles now."
As it to prove that The Rumour reunion is not some easy exercise in nostalgia, Parker says he has already written another batch of songs for the band which he is eager to record as soon as possible. "Being an artist is a stream," he says. "I'm not living in yesterday."
Graham Parker & The Rumour play O2 ABC, Glasgow, on 23 October. Three Chords Good is out now on Primary Wave Records.