Can a vectorized computation always be parallelized?
If there are dependencies between loop iterations, then the loop cannot be parallelized.  Depending upon the precise definition of "vectorized" you have in mind, your definition might imply that there are no dependencies -- or it might not.

Some examples:

foreach i in S:
    B[i] = f(A[i])


is parallelizable if f is a pure function and A,B don't alias (have no overlap).  (But it is in general not parallelizable if f writes to global state or performs any kind of I/O.)  More generally

foreach i in S:
    B[i] = f(i, x, y, ..)


is parallelizable if f is pure and does not access B (for reading or writing) and none of the arguments alias B.  However,

foreach i in S:
    B[i] = B[A[i]]


is not parallelizable, as there are dependencies between loop iterations.  (At least, it is not trivially parallelizable, if A is not known at compile time.)

I've written using foreach syntax but you can translate each example to map or apply and the same comments apply.