The forward completed the full game while his counterpart was withdrawn after the hour mark.
At a Glance
Antoine Semenyo's long throws created goals for David Brooks and Justin 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.
Bournemouth forward Antoine Semenyo, reportedly close to a big-money transfer, was at the heart of the action, providing two assists and conceding a penalty in a dramatic 2-2 stalemate against 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.
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.
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.
A potential handball incident involving Gusto was quickly reviewed and not given by the video officials. 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.
The result means Chelsea have managed just one victory in their last seven league outings. Bournemouth's winless streak now stretches to ten matches across all competitions.
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