From the guidance notes for UK passport applications:
Customer chooses not to send us voluntary DNA evidence
If a customer chooses not to provide DNA evidence, you must make a decision on
the application based on a balance of probability, using documents and information
the customer provides. You must not draw any negative inferences about a
customer’s decision against offering DNA evidence.
https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1112989/DNA_testing_for_British_passport_applications_V7.pdf
Seems you can decline a DNA test and, instead, provide evidence to show that "on a balance of probabilities" you are the father.