Thistle dominated the first half of the Scottish Premiership clash at Firhill and opened the scoring through an Aaron Muirhead penalty.
Boyd then threw Killie a lifeline with the leveller early in the second half but neither side could find a second goal to kill off their opponents.
The result means Kilmarnock are yet to get off the mark in terms of victories this term.
Boyd said: "We came alive in the second half but it wasn't good enough.
"That is the last two or three weeks where it's taken us 45 minutes to get going.
"The second half was a bit better but it is still not good enough and everyone in the dressing room knows that.
"Their keeper pulled off a couple of great saves but that was about it.
"As a team, we need to get back to the training ground on Monday and start addressing the problem."
Boyd is struggling to put his finger on the reasons for their poor form as Killie prepare to face joint leaders Celtic next week.
He said: "I haven't a clue why we are not starting games well.
"We have started slowly and it isn't until the second half when we start getting at teams.
"We probably should have more points than we have but we don't and now we need to start preparing properly for the Celtic game."
Boyd defended his challenge on Aaron Taylor-Sinclair, which earned him a yellow card that the home fans clearly felt should have been a red.
He said: "I knew I didn't touch him and I said that to him when I got up but he was still limping about.
"It looked bad but I never touched him. That is part and parcel of football."
Substitute Christie Elliott could have claimed the win for Partick after hitting the post and then passing up another couple of decent opportunities late on.
He said: "I had a few chances. I thought the keeper got a touch where he pushed it onto the post.
"He made a great save from my second one although, if I caught it right, it could have gone in.
"And the last one was just an unbelievable tackle.
"We are disappointed we didn't win it.
"We have been the better team in every game we've played apart from Aberdeen. We have nothing to fear."