Introduction
The sources for these surnames are Appendix 1, "Textile professions from BLC, EFF, and IPM grouped by occupation and role listed by property and year," and Appendix 2, "Textile professions from the control group selected by occupation and role listed by property and year" of The Textile Industry in Essex in the late 12th and 13th Century by Michael Gervers.
While the given names and place names in the Appendix have been modernized, the surnames have been retained in their original forms. I have listed the surnames in alphabetical order, organized by their modern counterparts and with derivation notes. Following each name are the dates that the name was found. (I have not included duplicate instances; a specific spelling might have been found four times in 1254, but I have listed the date only once.)
Burrel, from Middle English borel, Old French burel 'reddish-brown'
Capellarius, from Latin capellarius 'hatter, capper
Capron, from Old French chaperon 'hood or cap worn by nobles'
Challener, from Middle English chaloun 'blanket'
Coifer, from Old French coifier 'maker or seller of coifs'
Custer, from Old French coustier 'maker of feather-beds or cushions'
Draper, from Old French drapier 'maker or seller of woollen cloth'
Dyer, from Old English d{e-}agere 'dyer' [2]
Fuller, from Old English fullere, Old French fouleor, foleur 'a fuller of cloth'
Hatter, 'a maker or seller of hats'
Hood, from Old English hod 'hood'
Lyngedraper, from Old English lin 'flax' and Anglo-French draper from Old French drapier 'maker or seller of woollen cloth'
Napier, from Old French napier, nappier, from napper 'table-cloth'
Parmenter, from Old French parmentier, parmetier 'tailor'
Quilter, from Old French cuilte, coilte 'a quilt'
Scissor, from Latin scissor 'cutter'
Taylor, from Old French tailleor, tailleur 'tailor'
Teller, from Old French telier 'maker or seler of linen cloth, a weaver'
Textor, from Latin textor 'weaver'
Tinctor, from Old French teinturier, tainterier 'dyer'
Weaver, from Old English wefan 'to weave, a weaver'
Webber, from Old English web 'web, weaver'
Of unknown derivation
Notes:
[1] This derives from the Latin coffa 'coif'
[2] The {e-} represents an e; with a slash over it.
[3] This is a Latin, or written, form of the surname; in everyday usage it was probably Parmentar like the other example.
[4] This is a feminine form of the surname.
[5] This could be from either Textor or Teyntor
Sources:Gervers, Michael, The Textile Industry in Essex in the
late 12th and 13th Century, Appendices 1 & 2.
http://www.utoronto.ca/deeds/pubs/doc3/app1.htm
http://www.utoronto.ca/deeds/pubs/doc3/app2.htm
Reaney, P. H., & R. M. Wilson, A Dictionary of English Surnames (London: Routledge, 1991; Oxford University Press, 1995).