Father in Paris dig into last pie, showing lineage (8)
I believe the answer is:
pedigree
'lineage' is the definition.
(pedigree is a kind of lineage)
'father in paris dig into last pie' is the wordplay.
'father in paris' becomes 'pere' (I've seen this in another clue).
'into' means one lot of letters goes inside another.
'last' indicates one should take the final letters.
The final letter of 'pie' is 'e'.
'pere' going around 'dig' is 'pedigre'.
'pedigre'+'e'='PEDIGREE'
'showing' acts as a link.
(Other definitions for pedigree that I've seen before include "(Pure) ancestry" , "Pure lineage" , "Blood line" , "Line of descent, family tree" , "Ancestral line" .)