Yes, I can see a definite parallel.
Herodians were mentored by leading scholars in the ways of various Jewish sects and even identified with them to some extent. Josephus says he spent part of his youth as a disciple of Banus, who was probably an Essene or proto-Christian or both. However, after reaching adulthood and being appointed to positions of authority, he distanced himself from sectarian affiliations, or so he says.
Paul on the other hand persists in calling himself a Pharisee long after he abandoned the principles of that sect. It isn't clear however if Pharisees would ever have fully accepted him as one of their own. The legitimacy of Herod's family to rule the Jews was in general rejected, and especially by Pharisees.
Ultimately Paul (and no doubt other Herodians) became frustrated with the Jewish mentality of separatism. Paul must have convinced Nero that he could better integrate Jews and other groups by offering them a new cult, one that was comparable to the Serapis faith invented by the Ptolemies, but updated for the modern Roman world. Direct references to the Herodian Jesus were removed in this Gospel. The concept was popular enough, but not well received by the Jews.
So, Paul decided he would use the Roman government to punish and even kill those Jews who would not convert and accept his leadership. It was a doctrine of "believe, obey, or be damned". They still refused so their house (like that of the Pythagoreans) was burned down with them still inside it.
© Charles N. Pope, US Library of Congress. All rights reserved.