An alternate source suggests Joseph was the son of Tristram Manning and Joanna Drake.
Perhaps there were two. It appears he married Martha Drake (a cousin?).
Name Joseph Manning Gender Male Birth Date 29 Dec 1733 Birthplace PISCATAWAY,MIDDLESEX,NEW JERSEY Father's Name Isaac Manning Mother's Name Catherine
https://www.findagrave.com/memorial/86901456/william-manning
This person was created through the import of Watkins.ged on 04 April 2011. The following data was included in the gedcom. You may wish to edit it for readability.
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Categories: Estimated Birth Date | Uncertain Family