Primary source which mentions that Sir Laurence Aylmer's father was called Thomas Aylmer:
'gentleman Thomas Aylmer and Laurence Aylmer. his son'D/P 32/25/36Deed: Grant Thomas Laurence and John Terlyng of Harlow to Edward Buggys and Robert Buggys his son, gentleman Thomas Aylmer and Laurence Aylmer. his son, John Wright and Thomas Wright his son, John Spycer and William Spycer his son, Thomas Lawrence son of Thomas Lawrence, Thomas Lawrence son of Robert Lawrence, Robert Brangwyn, Lawrence Haver, George Tanbridge, Thomas Patryk, Henry Grene the younger, Richard Waylett, Thomas Herdy and John Thurgood All the lands in Great Parndon with which they and John Symond, Robt. Tyars, Thomas Laurence the elder, John Weylett the elder, Thomas Sampson, Hugh Dobbis, [Jo]hn Grene, Henry Algore of Great Parndon, William Clerk, citizen and s[alt]er of London, and John Algore, all now dead, were enfeoffed by Thomas C[lyfford], William [Sandes and Reginald Sandes, gen]tlemen Note: this charter is eaten away in the left centre, and one of the two seals is gone deed: Grant Thomas Laurence and John Terlyng of Harlow to Edward Buggys and Robert Buggys his son, gentleman Thomas Aylmer and Laurence Aylmer. his son, John Wright and Thomas Wright his son, John Spycer and William Spycer his son, Thomas Lawrence son of Thomas Lawrence, Thomas Lawrence son of Robert Lawrence, Robert Brangwyn, Lawrence Haver, George Tanbridge, Thomas Patryk, Henry Grene the younger, Richard Waylett, Thomas Herdy and John Thurgood All the lands in Great Parndon with which they and John Symond, Robt. Tyars, Thomas Laurence the elder, John Weylett the elder, Thomas Sampson, Hugh Dobbis, [Jo]hn Grene, Henry Algore of Great Parndon, William Clerk, citizen and s[alt]er of London, and John Algore, all now dead, were enfeoffed by Thomas C[lyfford], William [Sandes and Reginald Sandes, gen]tlemen Note: this charter is eaten away in the left centre, and one of the two seals is gone
Level: Item
Dates of creation 4 July 1513
Scope and content Grant
Thomas Laurence and John Terlyng of Harlow to Edward Buggys and Robert Buggys his son, gentleman Thomas Aylmer and Laurence Aylmer. his son, John Wright and Thomas Wright his son, John Spycer and William Spycer his son, Thomas Lawrence son of Thomas Lawrence, Thomas Lawrence son of Robert Lawrence, Robert Brangwyn, Lawrence Haver, George Tanbridge, Thomas Patryk, Henry Grene the younger, Richard Waylett, Thomas Herdy and John Thurgood
All the lands in Great Parndon with which they and John Symond, Robt. Tyars, Thomas Laurence the elder, John Weylett the elder, Thomas Sampson, Hugh Dobbis, [Jo]hn Grene, Henry Algore of Great Parndon, William Clerk, citizen and s[alt]er of London, and John Algore, all now dead, were enfeoffed by Thomas C[lyfford], William [Sandes and Reginald Sandes, gen]tlemen
Note: this charter is eaten away in the left centre, and one of the two seals is gone
https://secureweb1.essexcc.gov.uk/SeaxPam/ViewCatalogue.aspx?ID=29990http://seax.essexcc.gov.uk/Result_Details.aspx?DocID=30194The earliest reference I can find to 'Lawrence Aylmer, Draper' being the 'sonne of Thomas Aylmer, of Ellesnam in Essex' is in the book
The Survey of London: Contayning the Orignall, Increase, Moderne Estate, and Government of that City by John Stow from 1633.
Dr. Fuller refers to him in his book
in The history of the worthies of England by Thomas Fuller as 'Laurence Aylmer, son of Thomas Aylmer, of Allesham, Draper, 1507.'
The Boyds Inhabitants of London Image at FindMyPast describes him as 'Thomas Aylmer of Elsham Essex'.
https://search.findmypast.co.uk/record?id=GBOR/BIL/SOG31/0209&parentid=GBPRS/BIL/00179492