Option 1 -> "while" indicates contrast, but here we need to explain what the parents' wishes were, not contrast them.
Option 2 -> "so that" expresses purpose, which doesn't fit the sentence structure or the relationship between the clauses.
Option 3 -> "as" means "because/since" and provides the reason or explanation for the parents' wishes mentioned in the first clause.
Option 4 -> "and" simply adds information but doesn't show the causal/explanatory relationship needed here.
Hence, Option 3: as -> The connector "as" perfectly introduces the explanation of what the parents' wishes actually were—they wanted him to become an engineer instead of studying medicine. It establishes the causal relationship between the two clauses. -> correct