Stephan Lichsteiner and Arturo Vidal on their debuts alongside Simone Pepe and Claudio Marchisio scored the goals to hand the Old Lady of Turin the ideal start.
Former Juve forward Sebastian Giovinco scored an injury-time consolation from the spot after Paolo De Ceglie had been sent off for bringing him down in the box.
Having spent heavily in the summer on a number of recruits, Juve handed debuts to Andrea Pirlo, Lichsteiner and Emanuele Giaccherini whilst Mirko Vucinic and Vidal both came off the bench to also make their first appearances in the black and white shirt.
Despite the convincing victory, Conte was not an entirely satisfied man.
“In the first half there was a valid goal ruled out and we were denied a stonewall penalty,” complained Conte, who enjoyed a highly successful playing career with Juventus.
“At half-time I told the lads that 1-0 wasn’t a fair reflection of the game, which was partly our fault and partly due to other factors.
“Otherwise it was a good debut. Having said that I don’t want to con anyone, we weren’t perfect and there’s still a lot of work to do.
“You can’t concede a penalty and a red card at 4-0. We also have to work on this even if the attack, and especially the defence, did really well.”
Even though Juve’s new 41,000 capacity stadium wasn’t quite full, there was a scorching atmosphere inside the ground for its first official match.
And the hosts didn’t disappoint, coming out of the traps with energy and desire.
They were ahead on 16 minutes after a moment of genius from Pirlo, who scooped the ball over a static backline for right-back Lichsteiner to run onto and take around Antonio Mirante before slotting into the empty net.
Moments later Mirante had to be alert to block a shot from Alessandro Matri with his feet.
On 39 minutes Juve had the ball in the net again as Matri headed in from barely a yard out, but it was ruled out for offside.
Although Matri may well have been onside when Andrea Barzagli headed the ball back across goal, two other Juve players in close attention weren’t.
Parma showed nothing going forward in the first period with Giovinco particularly frustrating as he repeatedly tried to do too much on his own, losing the ball every time.
The second half followed a similar pattern to the first as Alessandro Del Piero released Matri into a gaping hole in the middle of the Parma defence but the forward’s shot struck the upright before rolling agonisingly across the line and away to safety.
The killer second came only four minutes later as Del Piero fed the ball wide to Pepe who thrashed a shot across Mirante and inside the far post from 12 yards.
Chile midfielder Vidal then came on to score the third with an acrobatic volley from just outside the area.
The fourth was simply sublime as Pirlo played another sumptuous chipped pass over the top and Marchisio ran past ball-watching defenders to brilliantly flick the ball over Mirante on the stretch.
Parma did get on the scoresheet in injury time as De Ceglie brought down Giovinco, who picked himself up to score the spot-kick.
Napoli started their campaign with a 3-1 victory at 10-man Cesena on Saturday.
Ezequiel Lavezzi gave them an early lead before Roberto Guana equalised in the first half.
Cesena had Yohan Benalouane sent off soon after the restart, after which goals from Hugo Campagnaro and Marek Hamsik sealed victory.
On Friday, in the Serie A kick-off, champions AC Milan were held to a 2-2 draw at home to Lazio.
New signings Miroslav Klose and Djibril Cisse got the visitors off to a flying start before goals from Zlatan Ibrahimovic and Antonio Cassano restored parity.