Yes, King Prasanajit who was Lord Rama's ancestor was also Lord Parashurama's maternal grandfather. So Lord Parashurama has links to Ikshavaku lineage.
From Chapter VII - Lineage of Puruvasas and Jahnu of Vishnu Purana,
Jamadagni married Reṇukā, the daughter of Reṇū, of the family of Ikṣvāku, and had by her the destroyer of the Kṣatriya race, Paraśurāma, ...
A partial answer
Apart from Aditya Hridayam and Bala and Atibala, the following are coming to my mind now. I will update later with some more, if found.
Sutra/Sloka stated by Sri Rama
सकृद् एव प्रपन्नाय तव अस्मि इति च याचते ||६-१८-३३ अभयम् सर्व भूतेभ्यो
ददामि एतद् व्रतम् मम |
"He who seeks refuge in me just once, telling me that I am yours', I