"But he became angry and was not willing to go in; and his father came out and began pleading with him” (NAS)
Here are the several different ways parakaleo is translated in Luke 15:28: “plead(ing/ed)” (NIV, TNIV, NKJV, HCSB, NAS, NRSV), “begged” (NLT), “entreat(ed/ing)” (ASV, ESV, RSV, YLT), and “request” (BBE).
Now, I should add that the Bible in Basic English (BBE) translated the phrase “pleaded with him” as “made a request to him to come in,” but this should be (mostly) written off as a typical expansion by the BBE without justification in the Greek words themselves, but by the overall context. This leads into why we can know that parakaleo by itself in this verse does not convey the idea of "coming in" or "invitation". Luke 15:28a says that the brother was unwilling to “go in.” Therefore, the translators of the BBE figure that the “going in” was the content of the father’s pleading/begging/entreaty. So, parakaleo simply conveys the idea of a request, an urgent request, maybe even begging. But no concept of “inviting” or “coming” is communicated by parakaleo in Luke 15:28.