The on-loan Chelsea ace fired home in the 70th minute at Craven Cottage but the hosts were unable to find an equaliser.
Loftus-Cheek's performance served as one major positive for his manager to take from the encounter, with the 24-year-old also able to win a penalty in his 32-minute cameo.
An Achilles injury in May last year halted the development of the England international and he has been used sparingly by Parker in November, but his boss hinted he is ready to unleash the Chelsea academy graduate.
"I thought he was brilliant," the Fulham manager said of Loftus-Cheek.
"We have worked really hard with Ruben in terms of the information I am trying to give him and him as a player and how he can have a big impact this year.
"Sometimes I feel like Ruben has a tendency for games to drift him by but today was everything we have spoke about with him.
"He came on, made a difference and looked every bit of a top, top player. He got on the ball, was brave, scored a goal and was dynamic and if you get him in between the lines, with confidence, he has that raw ability.
"He will be big for us this year. He needs to keep working and listening to what we are saying to him, but no doubt he will keep improving."
Loftus-Cheek will hope his goal can earn him a start away to Leicester on Saturday, which is the beginning of a daunting run of fixtures for the newly-promoted side.
Fulham have lost seven out of nine games in the Premier League and after a trip to the King Power Stadium at the weekend, take on Manchester City and Liverpool in consecutive matches at the start of December.
Parker added: "We can lose football matches and I will be judged on that, I understand that, but I don't think anyone will ever point the finger or tarnish us by saying the players won't play for me or this football club or show that.
"Every time they step on the field they will keep learning, they will make mistakes and the gulf at times is big with the quality of players and fixtures coming up, there is no doubt.
"But with endeavour and a work ethic to keep improving as a team, we will go in the right direction."
Everton boss Carlo Ancelotti was pleased to see his side return to winning ways after a four-match run without a victory.
The Italian described the spell as a "difficult period" but helped to move away from that with a change of formation in London from his usual 4-3-3.
Ancelotti warned: "I don't think it is a winning system because there is no winning system, but we tried to be more compact in the middle of defence with three defenders.
"Anyway we have to work on this. I think this system gave us the possibility to use more the width of the pitch and to put crosses in and with (Dominic) Calvert-Lewin and Richarlison, we can be dangerous."
Nigeria international Iwobi has been in and out of the team since he moved to Merseyside from Arsenal last summer, but Ancelotti was delighted with his display out of position.
The 61-year-old added: "He has the quality to play there because he is physically strong and can go up and down.
"He has quality one-on-one, can put in a lot of crosses and in the first half he did really well. I am happy for his performance."