Welcome to the Wefugees community and sorry for the wait due to the Holidays!
Yes, you are able to marry your Palestinian boyfriend in Greece- one of the core requirements to get married is that he is there legally, which applies to him, if he has been granted asylum. Since you are a UK-citizen, you can apply for family reunion once you two are married, which will give your partner the right to a residence permit to live with you.
It is important to check which documents you will need in order to get married in Greece and which documents the UK needs to accept your marriage. Also check the requirements for applying for family reunion: https://www.gov.uk/settlement-refugee-or-humanitarian-protection/family-reunion
There have been a few threads on our platform already where details about marriage in Greece have been discussed:
Can a refugee get married in Greece with an EU citizen?
I am British, can me and my Syrian husband, who has Asylum in Greece, move to England together eventually?
Also have a look at this page from Refugee Info on Getting Married in Greece (it states which documents you need): https://www.refugee.info/greece/marriage-in-greece/getting-married-in-greece/?language=en
I hope these sources help to answer your question.
Please feel free to reach out to us again if you need any further assistance!
All the best,