The resolution is simple: Kashyapa is a Chiranjeevi, so he was alive before the time of Narasimha, and he continued to be alive in the time of Parashurama.
Here is what the Harivamsa says:
In the course of time the eternal Kashyapa begat all created beings on Daksha's daughters.
And this chapter of the Anushasana Parva of the Mahabharata shows that Kashyapa has lived for at least 30,000 celestial years:
Once on a time, O king, a ruler of the name of Anga desired to give away the whole earth as sacrificial present unto the Brahmanas. At this, the earth became filled with anxiety. 'I am the daughter of Brahman. I hold all creatures. Having obtained me, alas, why does this foremost of kings wish to give me away unto the Brahmanas? Abandoning my character as the soil, I shall now repair to the presence of my sire. Let this king with all his kingdom meet with destruction? Arrived at this conclusion, she departed for the region of Brahman The Rishi Kasyapa, beholding goddess Earth on the point of departing, himself immediately entered the visible embodiment of the goddess, casting off his own body, by the aid of Yoga. The earth thus penetrated by the spirit of Kasyapa, grew in prosperity and became full of all kinds of vegetable produce. Indeed, O king for the time that Kasyapa pervaded the earth, Righteousness became foremost everywhere and all fears ceased. In this way, O king, the earth remained penetrated by the spirit of Kasyapa for thirty thousand celestial years, fully alive to all those functions which it used to discharge while it was penetrated by the spirit of Brahman's daughter. Upon the expiry of this period, the goddess returned from the region of Brahman and arrived here bowed unto Kasyapa and from that time became the daughter of that Rishi, Kasyapa is a Brahmana. Even this was the feat, O king, that a Brahmana did. Tell me the name of the Kshatriya who can be held to be superior to Kasyapa!
As you can see in my answer here, one celestial year is 360 human years, so at the very least Kashyapa has lived for 10.8 million years!
And by the way, it's not like Kashyapa was involved with Varaha and Narasimha, was absent during the Vamana Avatara, and then suddenly reappeared during the time of Parashurama. On the contrary, Vamana himself was the son of Kashyapa and Aditi as described in this chapter of the Srimad Bhagavatam.
And Kashyapa was even involved in the Mahabharata! He's mentioned as one the sages that Yudishthira will meet in his Tirtha Yatra in this chapter of the Vana Parva of the Mahabharata:
Those foremost of Rishis, Valmiki, and Kasyapa, and Atreya, and Kundajathara, and Viswamitra, and Gautama, and Asita, and Devala, and Markandeya, and Galava, and Bharadwaja, and Vasishtha, and the Muni Uddalaka, and Saunaka with his son, and Vyasa, that best of ascetics, and Durvasas, that foremost of Munis, and Javali of great austerities--all these illustrious Rishis endued with wealth of asceticism, are staying in expectation of thee. With these, O mighty king, do thou meet by visiting these tirthas.
This makes clear the longevity of Kashyapa.