Palace beat Southampton 3-0 on Saturday for their first win since a 3-2 victory over Sunderland on 24 September.
Pardew, 55, has been in charge since January 2015, but had faced increased speculation about his job recently.
"I'm pleased for the club and it was important we won today. The discipline was good," said Pardew.
Palace had lost their previous six matches with Pardew, who took the club to last season's FA Cup final at Wembley, describing the run as a "mini-crisis".
But two goals from Christian Benteke and one from James Tomkins gave the Eagles a victory over Southampton to move up to 13th in the Premier League table.
After the third goal, Pardew gave a thumbs-up gesture to Parish, who responded in a similar fashion.
"We have a lot of serious investors at the club who perhaps don't know a lot about football so the chairman has been defending me this week, so I just wanted to say thanks," added Pardew, who has also managed Reading, West Ham, Charlton, Southampton and Newcastle.
"When the pressure racks up in the Premier League it's hard but we got the balance right and we had some good attacking play.
"When you've been manager at Newcastle and the whole town lives and breathes football, you have to be able to cope with pressure.
"The pressure sometimes there was intense. It wasn't as bad this week."
Crystal Palace manager Alan Pardew thanked club chairman Steve Parish for his support after the Eagles won for the first time in eight matches.