The entire story of Baldur's Gate 3 will begin with your character searching for a way to remove the parasite from their brain through various means. From hunting down healers to making deals with devils, your character can explore a wide variety of options in order to try and rid themselves of this parasitic brain bug.
With so many options available to you, you wouldn't be alone in trying to figure out which option is the best one, both for your character and the world. Unfortunately, that question is still pretty tough to answer.
Seek Out Omeluum In The Underdark
At the time of this writing, there is no option that will completely remove the parasite from your character. However, there is one option that can give you the best case scenario as you continue looking for an answer in later Acts.
This option is working with Omeluum, who can be found in the Underdark area of the game. Now, this option may be surprising as the best, as Omeluum is a Mindflayer, one of the creatures who planted the parasite in your brain in the first place. However, Omeluum has been able to break away from the Elder Brain who controls their kind.
Now, Omeluum is dedicating their life to helping the life of any creature living in the Underdark and even trying to develop a type of food for Mindflayers that doesn't involve eating the brains of others. So, they're trying to do better and deserve to be trusted, at least somewhat.
In working with them, Omeluum won't be able to completely remove the parasite, unfortunately, but they will be able to give you the Ring of Psionic Protection. This ring will prevent any more damage being done by the parasite in your head as you continue looking for a real solution to rid your body of it completely.
Of course, this isn't as good as an outright cure, but this is your best option by the end of Act 1 as you continue searching the extensive game to track down the best way to do this.