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The former Celtic star grabbed his first Hammers goal as Parker's reign began with a rollercoaster after replacing Nuno on Monday.
West Ham looked dead and buried when Cardiff raced into a 3-0 lead, leaving their new boss furious with what he had witnessed.
And Engels admitted Parker let his players know exactly what he thought before his half-time blast helped spark an incredible fightback.
"Everybody was pretty angry," said Engels. "The gaffer as well, he was really angry.
"But everybody just reflected on themselves, looked in the mirror and said, 'We cannot lose this game.'
"We showed great mentality and that's what we needed to do."
West Ham responded with three goals after the break to salvage a point, with £22million Belgian international Engels netting his side's second.
The 23-year-old also caught the eye with his attacking intent as the Hammers pushed forward in search of an unlikely comeback.
Reports in England have suggested senior figures at the London Stadium are keen to see Engels deployed further up the pitch following his move from Celtic.
And he offered an immediate demonstration of what he can bring in advanced areas, making his mark on the scoresheet as Parker's decision to unleash him paid off.