Here are two References that list John as Henry's father.
Suffolk manorial families: being the county visitations and other pedigrees ... edited by Joseph James Muskett
3. Adam Winthrop 3d son; b. 10 Aug. 1548; of Edwardstone & after of Groton, Esq.; lawyer and County Magistrate. For his Diary see "Life and Letters of John Winthrop;" tomb in Groton churchyard 28 March 1623. mar. Anne, dau. & co-heir of Henry Browne of Edwardstone in co. Suffolk, clothier; 2d wife; m. 20 Feb. 1579; d. 19 April 1620. See BROWNE.
1st Gen. John Browne of Wigham in com. Essex; Will Arch. Colchester, 9 March 1562; probate 19 April 1563; names Elizabeth Raven, his daughter, and Margery, Anne, Joan and Anne, the four daughters of Henry Browne his son. ch: Henry Browne,; Elizabeth Browne, living 1596, sytled "my dau'r Raven" in her father's will. mar. John Raven, 35 Eliz., devisee of John Browne, 1562.; Thomas Browne. His children lving 1596; eldest son, 1562. mar. Elizabeth, devisee of John Browne, 1562.
2nd Gen. Henry Browne of Edwardston, co. Suff., and after of Milton in Prittlewell, co. Essex, Yeoman. Will P.C.C. 2 Cobham, 23 June, 35 Eliz.; prb't 27 Jan. 1596; ob. aet. 76; bur. in Prittlewell Church; Lands in Posted, Boxford, Edwardston, &c. mar. Agnes Browne, the mother of Anne Winthrop, died 17 Dec. a? 1590. Adam Winthrop's Diary. ch: Anne, da. and coh. 35 Eliz., devisee with her sister Ann Suelling, in her father's will. mar. Adam Winthrop ....; Joane Browne, da. and coh. named 35 Eliz. in her father's will. Her Will Arch. Suff. is dated 5 Sep. 1597. mar. William Hilles, of Holton, co. Suff. 2 Aug. 39 Eliz; prob't 20 Sept. 1597; names 1597 in Adam Winthrop's Diary. See Hilles.; Anne Browne, da. and coh, Anne, d. of Browne of Edwardstone in com. Suff. mar. John Snelling. - Harl. MS. 1560. mar. John Snelling of Boxford, Clothier, cousin to Wm. Ames, the Puritan Divine; Will P.C.C. 48 Meade, 26 Jan'y 1617. His 2d wife was Judith Choppin, Inq. p.m. 15 Ja'n.; Margerie Browne, da. and coh., wife of Mr. Weston, 35 Eliz., named in will of John Snelling, 1617; no issue by 2nd mar.; dead apparently in 1608. mar. ___ Speade, 1st husband, mar. Roger Weston, Vicar of Wormingford co, Essex, 2d husb'd. died 2 Dec. 1608; Will, Lend. Consist. 5 Feb, 1605; cousin of D'r Duke to be executor. Henry Browne mar. Margaret, living 1596, 2nd wife.
Evidences of the Winthrops of Groton, co. Suffolk, England, and of families in and near that county, with whom they intermarried (1894)