Although I cannot name Levi's father, here is some information that might help to further your research.
The marriage enty for Levi on p. 161 of West Bridgewater VR has:
"PACKARD ... Levi T., 22, shoemaker, s. Lucy "(now, Lucy Carr.)," and Fidelia Ann Higgins, 18, b. Starks, Me., d. Aaron and Apphia, June 30, 1849."
In the state copy of the West Bridgewater death records, vol. 302, p. 310, the death record of Levi T. Packard shows a death date of 19-May-1878, a married man of illegitimate birth, aged 50 y. 8 m. 19 d., died of apoplexy, a shoemaker, born at West Bridgewater, son of Lucy Packard.
Since we know that his mother later married a Carr, I checked the state copies of death registers for Plymouth Co. towns looking for a Lucy Carr. The most promising entry was in vol. 266, p. 330, for a Lucy Carr who died in West Bridgewater January [torn] 1874, a widow, aged 88 y. 2 m. 5 d., died of old age, a housewife, born at West Bridgewater, names of parents unknown. This gives a date of birth of about Nov-1785. This Lucy seems rather old to be Levi's mother, though.
Of course there was no town of West Bridgewater in 1785, as it was not incorporated until 1822, but we can assume that what was meant was the west parish of Bridgewater.
I have not been able to find a birth record for Levi or Lucy. I do find that Lucy Packard married Reading Carr 16-Oct-1836 [West Bridgewater VR 161].