Saints climbed within five points of the promotion play-off spot with a win over Palace ahead of games against Newcastle United, Bristol City and Birmingham.
All three of Southampton FC‘s next opponents are vying for a place in the top three, with extra places for promotion to the WSL available this season.
Parker, in his first season at the club, wants Saints to remain in the promotion race for as long as possible, and knows the next three games are crucial.
“It felt great to beat Crystal Palace. We know that last week’s performance against Sunderland is not where we want to be,” said Parker.
“The game plan was executed perfectly. We know that in the next run of fixtures that we have, we are playing against four really good teams.
“It continues on the weekend when we have Newcastle, after that we have Bristol City, and then Birmingham. It’s a really tough run.
“It’s massive to get three points at the start of that. It’s a huge lift, and hopefully this game can help get people back at St Mary’s.
“Hopefully they were entertained, even in the moments when we were under pressure. The fans were outstanding in those moments as well.
“When we were pressing, they were right behind us. I hope they enjoyed themselves. We really appreciate what they gave us against Crystal Palace.
“Hopefully, the belief will grow. The players all believe in what they can do; we’ve just been a little inconsistent in moments.”
Ellie Brazil, who scored three against Palace, echoed her manager’s sentiments and insisted Saints must maintain a high level of performance over a stretch of games.
She also hopes Saints can continue making St Mary’s a difficult place for away sides to play as they hunt promotion to the top tier.
Hat-trick hero Brazil added: “We also need to take the bits that didn’t go too well against Crystal Palace. We need to own those and take accountability.
“Hopefully, with that, we get even better. Don’t just focus on all of the good stuff. Yes, we need to praise ourselves a little bit.
“We’ve got some games against really big teams coming up, teams that play really good football.
“We are going to have to do exactly what we did against Palace in the next four, five, six games. It should be fun if it is anything like that game.
“We are making St Mary’s a fortress now. It’s not easy for teams to come here, and that’s what we need. You can’t come here and have an easy game.”