Both words (each and apiece) are needed here, as the intended meaning is that each parent gave $12000 to each aide’s family member. However, the sentence as published (which is as shown in question) is jumbled compared to what one normally sees; the each is differently placed. The sentence could read,

...it was disclosed that each of Mr. Ensign’s parents gave $12,000 apiece to the four members of the aide’s family...

Both words (each and apiece) are needed here, as the intended meaning is that each parent gave $12000 to each aide’s family member. However, the sentence as published (which is as shown in question) is jumbled compared to what one normally sees; the each is differently placed, as noted below. The sentence could read,

...it was disclosed that each of Mr. Ensign’s parents gave $12,000 apiece to the four members of the aide’s family...

The published sentence's structure is "...that of X each gave...", which means "...considering X, each of X gave...", which is equivalent to "...each of X gave...". The author's intent may have been to have each and apiece close together to account for the arithmetic of getting $96000 from $12000: two parents times four recipients times $12000.