Material in petition in support of parent (6)
I believe the answer is:
damask
'material' is the definition.
(I know that damask is a type of material)
'petition in support of parent' is the wordplay.
'petition' becomes 'ask' (I've seen this in another clue).
'in support of' is a charade indicator (letters next to each other) (in a down clue, some letters hold others up).
'parent' becomes 'dam' (term for the female parent of an animal).
'ask' put after 'dam' is 'DAMASK'.
'in' is the link.
(Other definitions for damask that I've seen before include "Fine linen with woven pattern" , "Silk or linen material woven with pattern on either side" , "Figured material" , "Patterned material" , "reversible material" .)