But surely 'his' family are now 'your' family now too if you're married to each other?
WTF, by that logic then the OP's DH should be dealing with her mums/sisters/dads/uncles birthdays? 
The ONLY way this works properly (from what i can tell) is that either social planning is entirely 1 person's responsibility, as long as they're willing and good at it... or it's every man/woman for themselves and each deals with each other's direct relatives.
At no point would i have been happy to take on the wifework of doing this for DH's family.
i mean, i have had a message frmo DH along the lines of "it's mum's birthday next week, can you pick up a card on your way home tonight when you walk past [card shop]" but that's literally it - he remembers it/arranges it/writes it/sends it. i'm not putting all of that into my brain too!
similarly, i've asked him to do similar for me before (but it's my responsibility to remember when parents and brothers birthdays are).
i was always shocked at families where all this is lumped on 1 person in cases where both parties work/roughly have the same amount of work or hobby time, it seems so old fashioned (and unfair)