First half goals from Kyel Reid and James Hanson put City in control of the second round tie before injury time efforts from substitutes Gary Liddle and Devante Cole wrapped up the win at the Coral Windows Stadium.
Parkinson said: "We knew this game was going to be a difficult one because we were aware of how much Chesham had been looking forward to this tie.
"This game was massively important for us as well though, and I thought we did the job well.
"We played some great football in the first half and scored two very well-taken goals.
"We were a bit untidy in our play at times in the second period but the back four has stood strong and we saw the job through, which was very important.
"I'm pleased for Kyel. He has two great feet and he has struck the goal well.
"'Evo's (Lee Evans) pass for Hans' (James Hanson) goal was top class, and James has finished well on his weaker foot.
"It was a great run from Lids (Gary Liddle) for his goal at the end and a good finish. He was unlucky not to be starting today (Sunday) but I just felt Evans and (Billy) Knott were the right two for us in midfield for the game.
"Devante is obviously frustrated not to be starting - you want players to be disappointed when they are not in the team - but he's got his goal and it will do him good."