Ravana appears in Ramayana only and Parasurama appears in Mahabharata only. Hence, their meeting is not possible.
The episode of Parasurama challenging Sri Rama appears in the 74th - 76th sargas of Bala Kanda.
On hearing the unregenerate and highly perfidious murdering of his
father, Parasurama rancorously extirpated Kshatriya-s as and when they
are born, that too not for one time, going around the earth... and on
getting the entire earth under his control he performed Vedic-ritual,
and at the end of that Vedic-ritual.
He gave all that earth to sage Kashyapa, a sage with divine soul and
with pious observances, as a ritualistic-generosity."
On hearing about the breakage of Shiva's longbow, then he promptly came there and challenged Sri Rama.
The Story of Parasurama also appears in Mahabharata, wherein it was mentioned that Drona obtained all his divine weapons from the former.
Drona heard that the illustrious Brahmana Jamadagnya, Rama the son of
Jamadagni, had expressed a desire of giving away all his wealth to
Brahmanas. Having heard of Rama's knowledge of arms and of his
celestial weapons also, Drona, who set his heart upon them as also
upon the knowledge of morality that Rama possessed, set out for the
Mahendra mountains.
Arrived at Mahendra, Drona approached Rama the son of Jamadagni,
worshipped Rama's feet, and told him that he came there with the
desire of obtaining his wealth.'
On hearing him, Rama said that he gave away his gold and whatever
other wealth he had to Brahmanas! He further stated that the earth
also, to the verge of the sea, decked with towns and cities, as with a
garland of flowers, was given unto Kasyapa. And, he added that he was
left with his body only and his various valuable weapons and that he
was prepared to give either his body or his weapons.
Drona prayed to give him all his weapons together with the mysteries
of hurling and recalling them and Rama gave all his weapons to Drona,
If we study these 2 stories in depth, we will find 2 contradictions.
1) If the Parasu Rama the son of Jamadagni extirpated Kshatriya-s many times by going around the earth, then lineage of both Ikshvaku's, to which Dasaratha belonged, and Nimi's, to which Janaka belonged to, must have been broken, as Parasu Rama might have eliminated Kshatriya-s of both lineages.
However, while describing the lineages of Ikshvaku's and Nimi's during the marriage ceremony of Sri Rama and Sita, Sage Valmiki did not mention about breakage of those lineages due to Parasu Rama's adventures.
2) It was mentioned in Mahabharata that Parasurama eliminated Arjuna (the Haihaya king), for revenging his father's death. However, Parasurama's encounter with Sri Rama was not mentioned here.
3) In Mahabharata it was mentioned that Drona obtained all his divine weapons from Parasu Rama.
Mahabharata followed Srimad Ramayana, as mention about Sri Rama can be found in Mahabharata but not about Pandavas in Srimad Ramayana. Many such instances can be quoted.
So if Parasu Rama distributed all his riches along with entire land on earth to Kashyapa and retired to Mt. Mahendra for practising ascesis, how can he still wait for Drona of Mahabharata era for imparting knowledge in divine weapons and then retire to Mt. Mahendra?
The answer is clear.
Parasu Rama existed only in Mahabharata era but not in Srimad Ramayana era.
