The way in which the Law 'entered' indicates the reason that the παραπτομα, paraptoma multiplied. (πλεοναζο, pleonatso, is not to merely 'increase'; it is to abound.)
παρεισερχομαι, pareiserchomai, is not just 'entered'. The prefix παρεισ, pareis, is a matter of 'being led in by one's side' (Liddel & Scott); 'to be at hand' 'to be present' (Thayer).
Examining the way in scripture that the four other verbs with the same prefix (namely παρεισαγο, παρεισακτος, παρεισδυνο and παρεισφερο) are used, it becomes clear that the idea is of an uninvited guest. The proof of this is too lengthy to report here, but it is clear from the four texts involved.
This witness was not invited in. It accompanied someone else, but it, itself, had no invitation. Israel followed Moses out of Egypt and through the Red Sea, but afterwards another witness joined Moses (as it were) at Sinai. Uninvited.
Once the Law witnessed to what men were doing, it became clear that the world was full of abundant paraptoma.
Ptoma is a corpse. Para is a matter of alignment.
It became clear that the state of humanity was as close to a corpse as it is possible to be without actually being in the grave.
It became clear, due to the uninvited witness of Law, that humanity was (in the words expressed by Paul in Ephesians 2:1) 'dead in trespasses and sins'.
It became clear, in fact that humanity did nothing but offend, for the Law, now in the world as an uninvited guest, witnessed to every deed, every word and (to those who were honest enough to admit it - see Romans 7) every desire, for the Law said, Thou shalt not covet.
Thus the Law witnessed to outward actions and it came right inside (if one was awakened to its true activity) and spoke to the inner man of the interactions within the spirit of humanity.
Thus did the Law 'enter' and thus did trespasses multiply.