Torn on this one OP.
Your comment was perhaps a bit harsh but sent in the heat of the moment. Were you all able to go to the party or was it not possible? Was the trip you wanted to take in place of seeing him at the party for his birthday?
We had a big falling out with my IL's once (more than once actually, but this specific time is sort of relevant to your story) and they were awful. The behaved spectacularly badly all through Christmas and New Year and besieged us with telephone calls for five weeks, four of which I was very ill for, and drove DH almost mad with guilt and anger.
It was so bad we kept our mobiles switched off because they were constantly ringing us to abuse us and I even turned off the ringer on the landline. They rang so often that on the odd times we switched the ringer back on the phone was either ringing as we did it or rang very soon afterwards. All hours of the day and night, phone calls and messages just filled with crying and shouting.
Eventually, when I felt better from my illness, I wrote to my IL's and told them that their behaviour had been awful and that they were driving their son away, which at the time was something I didn't want to happen.
FIL rang us later that day and demanded, in a very angry tone, to speak to "his son" and I said no (DH was out anyway) because I had written the letter and if they were going to have a go at him because of it they would be better of taking it out on me. FIL said "What I have to say to MY SON is between him and me NOT YOU" and put the phone down on me.
I rang back, told them that the next time they put the phone down on me would be the last time as I would have the numbers changed and not give them to PILs and that if FIL was intent on having another go at DH then that would be the moment they lost him for good because he was at breaking point.
I don't know where I got so brave but my point was, we are married and I wasn't having them exclude me under the banner of him being their son when he was also my husband. So I can see your SM's point when she decided to email you.
But from your side of things, I wrote the letter to them so I should have been the one they shouted at because of it (although really I don't think they should have shouted at all and actually after my phone call back to them they spent a few hours thinking about my threat and eventually rang back to apologise - to DH but not me) so I do think that if anyone was going to rant at you for your email then it should have been your father.
None of this is any use to you whatsoever but at least you know you're not the only one to send ranty correspondence to family and then bear the brunt of the reaction.
Sadly my PILs never got any better, only worse and now I never see them and DH sees them very rarely.
My only suggestion is to tell your dad that you love him, you were sad and disappointed that you wouldn't be able to see him for his birthday and you lashed out unfairly because you were hurt. Apologise to both of them and say you just want to be able to spend time with them both and your boys. Tell them you love your sister and wish that you could have as close a relationship with him as she does.
And hopefully that should help. I hope it all works out for you.