Unfortunately, we don't know what Richard thought about his father being disinterred, ritually executed and his head put on a spike. The main reason for this was that Richard went into voluntary exile to the continent in June of 1660 (Hutton, Restoration, p. 135). Oliver Cromwell's body was disinterred on January 28, 1661, hung on January 30, and the head placed on a spike sometime before February 5, 1661 (Diary of Samuel Pepys, 11 volumes, 2:24-31). By the time of the return of Charles to London and the violence towards his father's corpse, Richard was long removed from England. He would spend the next 20 years in exile, staying mainly in France, and potentially traveling to Italy.

The main problem is that the source record for Cromwell's time in Europe from 1660 to 1680 is incredibly thin. We have 5 letters of his that seem to date from the time of his exile, and none mention his father Oliver (Burn, "Correspondence of Richard Cromwell", EHR, 1898). In his later years and correspondence, he also does not seem to mention his father. Despite Richard living until 1712. Richard lived well enough drawing about £120 each year from his property, but kept largely to himself. He last saw his wife in 1660, and she died in 1676; Richard used pseudonyms for most of his life, changing his name so none would recognize him as the son of Oliver (ODNB, "Richard Cromwell" by Peter Gaunt).