A middle initial would really help narrow down results on this one, if available, given how common this name probably is.
Here are two John Parkinson listings I found from the 1920 and 1930 census with estimated birth years of 1920:
John Parkinson of Ridley, Delaware, Pennsylvania
Name: John Parkinson
Event: Census
Event Date: 1930
Event Place: Ridley, Delaware, Pennsylvania
Gender: Male
Age: 10
Marital Status: Single
Race: White
Birthplace: Pennsylvania
Estimated Birth Year: 1920
Immigration Year:
Relationship to Head of Household: Son
Father's Birthplace: Pennsylvania
Mother's Birthplace: Pennsylvania
Enumeration District Number: 0119
Family Number: 305
Sheet Number and Letter: 16A
Line Number: 2
NARA Publication: T626, roll 2032
Film Number: 2341766
Digital Folder Number: 4639363
Image Number: 00852
Household Gender Age Birthplace
Head Alfred Parkinson M 42 Pennsylvania
Wife Anna Parkinson F 38 Pennsylvania
Son Thomas Parkinson M 12 Pennsylvania
Son John Parkinson M 10 Pennsylvania
Daughter Jane Parkinson F 5 Pennsylvania
John W. Parkinson of Herkimer, New York
Name: John W Parkinson
Residence: , Herkimer, New York
Estimated Birth Year: 1920
Age: 0
Birthplace: Pennsylvania
Relationship to Head of Household: Son
Gender: Male
Race: White
Marital Status: Single
Father's Birthplace: Connecticut
Mother's Birthplace: Connecticut
Film Number: 1821115
Digital Folder Number: 4313447
Image Number: 00346
Sheet Number: 11
Household Gender Age Birthplace
SELF Richard E Parkinson M 29y Connecticut
WIFE Agnes C Parkinson F 24y Connecticut
SON Richard D Parkinson M 4y8m Connecticut
DAU Ruth E Parkinson F 3y2m Pennsylvania
SON John W Parkinson M 10m Pennsylvania
John Anderson M 26y Connecticut
Update: In the 1930 census, John W. Perkinson has an
estimated birth year of 1919.