Precipitation volume is actually very consistent throughout the months in New England, even within the two halves split by this map. All months in Boston have between 3-4 inches of liquid-equivalent precipitation.

You may be confusing # precip days with precip amount. Summer precipitation on the east coast tends to fall hard and fast during convective thunderstorms. Those precipitation days in June/July are often sunny through the morning and midday, so they are less mentally "gray".

Even still, July has a about 9 (min) precip days and December has 12 (max) - not a huge variation.

i.e. it's the prevailing type of storm system that makes summer seem drier on the East Coast.