Semenyo played the entire match as Palmer was substituted in the second half.
At a Glance
Two assists from Semenyo's long throw-ins led to strikes by Brooks and Kluivert.
Chelsea's goals came from Palmer and Fernandez during a chaotic opening 45 minutes.
Enzo Fernandez missed the best opportunity of a goalless second period.
In the midst of advanced transfer talks, Bournemouth's Antoine Semenyo delivered a mixed performance, setting up two goals and giving away a spot-kick in a thrilling draw with Chelsea.
The 25-year-old, who is in deep discussions over a potential multi-million pound move, nonetheless featured for his current side. His distinctive long throw-ins proved a potent weapon, leading to goals for teammates David Brooks and Justin Kluivert.
The deadlock was broken early when Semenyo's first long throw caused havoc, allowing Brooks to score from close range in the sixth minute.
The home side equalized and went ahead following a foul by Semenyo on Estevao in the box. Referee Sam Barrott pointed to the spot after consulting the Video Assistant Referee (VAR) monitor.
Palmer calmly dispatched the penalty, placing it perfectly into the corner even though the keeper dove the right way. Enzo Fernandez then extended Chelsea's lead, firing a superb shot into the top corner after 23 minutes.
Kluivert leveled the scores again shortly after, tapping home from another precise long throw delivered by Semenyo.
The visitors finished the first period on top, forcing Sanchez into several saves from their 14 first-half efforts.
At halftime, Chelsea's manager made two changes, introducing captain Reece James and Pedro Neto for Josh Acheampong and Alejandro Garnacho.
Early in the second half, a handball appeal against Malo Gusto was dismissed by the VAR official. Petrovic, once of Chelsea, made a fantastic stop to keep out Estevao's powerful drive from the edge of the area.
The best chance of the second half fell to Enzo Fernandez in the 77th minute, but he blazed his shot over the bar from inside the box.
Moments earlier, Palmer was substituted for Joao Pedro, a change that was met with disapproval from some home supporters. Although they enjoyed more possession late on, the hosts were unable to craft a winning goal.
In injury time, Bournemouth substitute Enes Unal volleyed over from 12 yards out after a cross from Amine Adli.
This draw extends Chelsea's poor run to only one win in their previous seven Premier League matches. Bournemouth's winless streak now stretches to ten matches across all competitions.
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