Parasite caught fish up on horse! (5)
I believe the answer is:
leech
'parasite' is the definition.
(I know that parasite can be written as leech)
'caught fish up on horse' is the wordplay.
'caught' becomes 'c' (cricket abbreviation).
'fish' becomes 'eel' (eel is a type of fish).
'up' is a reversal indicator (in down clue: letters go upwards).
'on' means one lot of letters go next to another (in a down clue, letters appear on others).
'horse' becomes 'H' (abbreviation as in horse power).
'c'+'eel'='ceel'
'ceel' in reverse letter order is 'leec'.
'leec'+'h'='LEECH'
(Other definitions for leech that I've seen before include "scrounger" , "Proverbial bloodsucker" , "Old physician" , "Old medical attendant" , "Archaic word for physician" .)