Semenyo played the entire match as Palmer was substituted in the second half.
Match Summary
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.
Despite being on the verge of a lucrative switch, the 25-year-old attacker started for his club. He utilized his long throw expertise to great effect, creating scoring opportunities that were finished by Brooks and Kluivert.
Brooks opened the scoring just six minutes in, converting from close quarters after a dangerous long throw from Semenyo.
Chelsea responded and then took the lead after Semenyo fouled Estevao Willian inside the penalty area. The match official awarded a penalty after a review by the Video Assistant Referee system.
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.
Chelsea created further opportunities, but Bournemouth ended the first half strongly, testing goalkeeper Robert Sanchez with numerous attempts.
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. Former Chelsea goalkeeper Djordje Petrovic then produced an excellent save to deny Estevao from distance.
Fernandez had the clearest opportunity to win it late on, but he skied his effort from a promising position inside the penalty area.
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.