A death in 1863, in the Lancaster Gazette (online newspaper archive) Saturday, November 28, 1863; pg. 5; Issue 4000
"Lancaster - On the 22nd inst [November] John Parkinson, sawyer, Dispensary, aged 63 years"
In 1861, this John was widowed, a joiner, living at 10 St George's Quay with married daughter Helen Rumley age 27, and his son Charles Parkinson age 20. All give place of birth Lancaster.
So he was born around 1800 - A cousin of your John perhaps? Or an older son even, if your John was widowed on marriage to Hannah Lamb?

Barbara