“Ultimately, results in recent months have placed us in a precarious league position and we felt a change is necessary to give us the best chance of retaining Premier League status.”
Under-21s boss Paddy McCarthy will take charge for Sunday’s visit to leaders Arsenal on Sunday (14:00 GMT) after he led training on Friday morning.
“Obviously it’s a massive moment for myself, but ultimately it’s not about me,” said McCarthy, who will work alongside U21s assistant Darren Powell.
“I’ll push all of that emotion to one side. Yes, it’s going to be a great opportunity for myself and Darren to manage at that level, but the focus is solely on setting the gameplan and implementing that on Sunday.”
Frenchman Vieira, 46, was appointed before the start of the 2021-22 season and led the Eagles to a 12th-placed finish, earning praise for an attacking style.
However, this season they have scored just 21 goals in 27 games – only Wolves, Everton and bottom-of-the-table Southampton have fewer.
Before their defeat by Brighton, Palace had gone three consecutive games without a shot on target.
No Premier League team has won fewer points in 2023 than Palace, who have taken five from a possible 33.
Three members of Vieira’s coaching staff – Osian Roberts, Kristian Wilson and Said Aigoun – have also left the club and the process of appointing a new manager is under way.
Just five points separate the Premier League’s bottom nine clubs heading into the final months of the season.
Their final 10 games of the season will see them face all eight of their relegation rivals.
Parish added: “Patrick’s impact since joining us in the summer of 2021 has been significant, and he is held in the highest regard by myself, and all of his colleagues.
“He led the team to a Wembley FA Cup semi-final and respectable 12th-placed finish last season playing some exciting football, which was a challenging and crucial campaign for the club given the changes we made to the squad prior to his arrival.
“Patrick has given his all to the club, and we all thank him and his team for their service.”
Vieira’s departure means there are no black managers in charge of a club in the top flight of men’s and women’s football in England.
In the wider Premier League, only six managers have been in charge for more than three years and nine have been in charge for fewer than 12 months. The league average stands at two years and five days.
There have been nine managerial sackings during the 2022-23 campaign – one fewer than last season’s total, which equalled the Premier League record.
Vincent Kompany, whose Burnley side lead the Championship by 13 points and play Manchester City in the FA Cup quarter-finals on Saturday, has questioned the decision to sack Vieira.
“It doesn’t sit well with me because if someone is underachieving in comparison to the resources you have then I think it is fair enough that a club makes a decision,” the 36-year-old said.
“But I think if you are overachieving, which is what they are doing, then I think whoever is making the decision has got to come forward and say ‘I made the decision because I think at the end of the season we are going to be safe because of that decision’.
“What you have done is taken out a manager who is overachieving, no matter which way you want to turn it. I think that is a problem in general in football.”
‘The threat of relegation proved too much’ – analysis
Football news reporter Alex Howell
The sacking of manager Patrick Vieira by Crystal Palace will not come as a surprise to those who have seen the Eagles struggle in 2023.
It was felt that Vieira was fighting for his job after the loss to Brighton on Wednesday, but he may have been given the Arsenal game as one last chance to show he could turn around a season that is turning into a relegation battle.
Goalscoring has been a huge issue – Palace have not found the net for the past four matches and have not scored twice in a game since December.
It is not yet clear who the club will bring in to replace Vieira but they will need to get the team scoring and quickly. The Eagles do not concede a lot but without scoring themselves, the threat of relegation gets bigger with every week that passes.