After Killing his mother Renuka, Parasurama asked some of his wishes to his father. One of the wish is that he(Parasurama) might not be affected by any sin for killing his mother. Along with this wish, Parasurama's father Jamadagni granted all wishes asked by Parasurama. Thus Parasurama is unaffected by the sin for killing his mother and hence there is no need to perform expiation for killing his mother.
It can be clarified from the following passage of the Mahabharata
"Akritavrana said, '........... At that very moment came in the eldest
of Jamadagni's sons, Rumanvan; and then, Sushena, and then, Vasu, and
likewise, Viswavasu. And the mighty saint directed them all one by one
to put an end to the life of their mother. They, however, were quite
confounded and lost heart. And they could not utter a single word.
Then he in ire cursed them. And on being cursed they lost their sense
and suddenly became like inanimate objects, and comparable in conduct
to beasts and birds. And then Rama, the slayer of hostile heroes, came
to the hermitage, last of all. Him the mighty-armed Jamadagni, of
great austerities, addressed, saying, 'Kill this wicked mother of
thine, without compunction, O my son.' Thereupon Rama immediately took
up an axe and therewith severed his mother's head. Then, O great king,
the wrath of Jamadagni of mighty soul, was at once appeased; and
well-pleased, he spake the following words, 'Thou hast, my boy,
performed at my bidding this difficult task, being versed in virtue.
Therefore, whatsoever wishes there may be in thy heart, I am ready to
grant them all. Do thou ask me.' Thereupon Rama solicited that his
mother might be restored to life, and that he might not be haunted by
the remembrance of this cruel deed and that he might not be affected
by any sin, and that his brothers might recover their former state,
and that he might be unrivalled on the field of battle, and that he
might obtain long life. And, O Bharata's son, Jamadagni, whose
penances were the most rigid, granted all those desires of his son...'
[SECTION CXVI, Tirtha-yatra Parva, Vana Parva, The Mahabharata]