You are not required to request a Social Security Number (SSN) on your Form I-485, and USCIS has stopped allowing SSN requests on Form I-765.
“I already applied for an SSN at the SSA office under my K-1 visa and am waiting for the card. On my I-485 application, should I select 'No' for the SSN request since I have a pending application? I'm worried that if I say 'No' and my SSA application is rejected, I won't be able to work even if my I-765 is approved.”
Summary
You do not need to worry about a duplicate filing error if a new request reaches the SSA. The agency instructs its technicians to process your pending application instead of the new one once they match your records.
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How pending applications and new requests interact
The official Form I-485 instructions state that you are not required to request an {"term": "ssn", "text": "SSN"} using the application.
For your Form I-765, USCIS has stopped allowing applicants for an {"term": "ead", "text": "Employment Authorization Document"} to request an SSN on the form.
If a new application reaches the SSA while you have a pending record, the agency instructs its technicians on how to handle it. If the technician determines that the pending record belongs to you, they must continue to process the pending application instead of the new application.
If your pending SSA application is rejected and you do not receive an SSN through your USCIS forms, you are required to submit an application in person at a local SSA office.
Legal sources
The rules governing SSN requests on USCIS forms and how the SSA handles pending records.
USCIS Form I-485 Instructions
You are not required to request a Social Security Number on Form I-485.
You are not required to request an SSN using this application.
USCIS Ends the Option to Request a Social Security Number When Applying for an EAD
USCIS no longer allows applicants to request an SSN on Form I-765.
U.S. Citizenship and Immigration Services (USCIS) has stopped allowing applicants for an employment authorization document (EAD) to request a Social Security Number (SSN) on their Form I-765 EAD application
SSA POMS RM 10205.700
SSA technicians are instructed to process a pending application instead of a new one if they determine both belong to the same applicant.
If you determine that the pending record belongs to the applicant, continue to process the pending application instead of the new application.
USCIS Ends the Option to Request a Social Security Number When Applying for an EAD
Foreign nationals who need an SSN generally must apply in person at an SSA office.
Foreign nationals seeking a Social Security Number or replacement Social Security card must generally appear in person at a local Social Security Administration office.
This is general guidance, not legal advice. Confirm these details with a licensed immigration attorney.
SettleKit is not a law firm. This information is for educational purposes and does not constitute legal advice.

