March 29, 2017 0307 GMT
Enrique confronted a television reporter working for Catalan station TV3 after the match, while his tactics were criticized by Barca midfielder Sergio Busquets, AFP reported.
Barcelona sports daily Sport's front cover said "this is not Barca", while Madrid-based AS's headline ran "total disaster”.
A former Real Madrid foe came back to haunt Barca as Angel di Maria scored twice. Further goals from Julian Draxler and Edison Cavani handed PSG the glorious European night it has craved since a takeover from Qatar Sports Investments six years ago.
The joint heaviest defeat in Enrique's three seasons in charge means Barcelona faces missing out on the quarterfinals for the first time since losing on away goals to Liverpool in the last 16 in 2006/07.
"It was a disastrous night for us in which we were clearly inferior," Enrique acknowledged after the game.
"I accept responsibility, but I would also like the same respect and personal treatment in the interviews when we win," Enrique said in a fiery interview with TV3.
Enrique's future at the Camp Nou beyond the current campaign was already in doubt as his contract expires at the end of the season. Barcelona also trails Real Madrid in the race for the Spanish title.
And Busquets's reaction in praising Enrique's counterpart Unai Emery for the tactical set up that caught the Catalans cold was telling.
"It wasn't an attitude problem, but a football one. They pressed more and were better tactically. They had a plan and carried it out as they wanted."
Sport described Barca as a "shipwreck without a coach".
"Barca fell in Paris without grandeur and without excuses," continued Sport's editorial.
"It was a tumultuous fall, from the first minute to the last...more than an accident, it manifested itself as an aggravation of many problems, some latent and others evident, that had already been shown up throughout the season."
Enrique's preference for Portuguese international Andre Gomes in midfield has been a particular bugbear for the press.
Gomes missed a huge chance to level with the score a 1-0 midway through the first-half.
Yet, humiliation in Paris laid bare broader failings in Barca's season on and off the field from the president down to the players.
Gomes was the most expensive of six summer recruits costing over €120 million ($127 million) designed to add greater depth and more youth to the squad.
Yet, none of them have made a significant impression with Barca still reliant on an ageing core of Gerard Pique, Andres Iniesta and Busquets alongside the start front three of Lionel Messi, Luis Suarez and Neymar.
Meanwhile, suitable replacements for the departed Dani Alves and Xavi Hernandez from the Barca sides that won trebles under Pep Guardiola in 2009 and in Enrique's first season in 2015 have yet to be found.
The most immediate order of business for president Josep Maria Bartomeu and his board is to tie down Messi with the five-time World Player of the Year just over a year away from becoming a free agent.
Yet, worryingly for Barca's future, Messi and Suarez also turn 30 this year.
The punishment dished out to Enrique was only further evidenced by the praise lavished on Emery in his homeland.
Emery guided Sevilla to Europa League glory in each of the past three seasons and once again proved his worth on the European stage despite having managed just one win over Barca in his previous 23 attempts with Sevilla and Valencia.
"The Sheikh signed him for this," said Madrid sports daily Marca.
"What Carlo Ancelotti and Laurent Blanc couldn't do, Unai has within his grasp — eliminating Messi's Barca from the European Cup."
Di Maria's metamorphosis
One month ago, Angel di Maria was an outcast on the Paris Saint-Germain substitutes' bench, but the striker was reborn in Tuesday’s demolition of Barcelona.
A brace by Messi's compatriot saw birthday boy di Maria hit near perfection in a way that he rarely managed at Manchester United or Real Madrid. Superstar Messi has rarely looked so poor.
Long-suffering PSG players and fans, so often at the receiving end against the Spanish giants, still could not believe the scoreline when they woke up on Wednesday.
"You saw 14 warriors out there, people who wanted to help each other. And when you play like that you have nights like that. It was fabulous," said midfielder Blaise Matuidi.
Di Maria hit a goal in each half on his 29th birthday and each time gave his traditional celebration, forming a heart with his hands.
But di Maria's heart has not really been in the PSG game this season. Last month, coach Unai Emery banished the second most expensive import into the French league — €63 million paid to United in 2015 — to the substitutes' bench.
Julian Draxler had arrived from Germany and Lucas was playing well. Being out in the January cold jolted di Maria into action.
Already against Bordeaux last Friday, he scored one in the 3-0 win that kept PSG in the French title race against Monaco.
PSG back Thomas Meunier said di Maria hated being on the bench and hailed di Maria's game against Bordeaux as "an Angel who has gone through a metamorphosis."
Di Maria said he had problems with the tactics brought in by Emery last year.
"Those who know me know that I like to run everywhere, a bit haphazard. He wants every player to take care of his own space and press from that position. It was a problem at the start, but I understood and now I like it," di Maria told the Spanish sports daily Marca after the Bordeaux game.
Di Maria forced Lucas out of the starting lineup for the Barcelona game and received a standing ovation from the Parc des Princes crowd when he left in the 61st minute.
He scored with a curling free kick and a second half shot, both attempts from outside the penalty area. Di Maria now has nine goals for the season and hopes it will set him up for a spectacular finale.
"It was a very special day. It's impossible to have a better performance and birthday. It was a wonderful display and fully deserved," he said on BeIn TV after the match.
"We controlled the game for 90 minutes. It was hard to control Neymar, Luis Suarez and Lionel Messi but we managed it in the end."