Both parts A and b have three digits multipliers. Why are there three partial products in part a and only two partial products in part b

There are three partial products in part a because both values, 273 and 346, have all numerical values, that aren't 0, so they can be multiplied and a value will be added.

There are only 2 partial products in part b because in 306, the tens place is worth 0 and anything times 0 = 0. There will be nothing to add for that value.