Good question! And one that's often been asked. And for myself, I've yet to find a fully satisfactory answer. Even from
Tolkien! My opinion blathers on like this...
... we know (from Appendix A) that the choice was extended to the children of
Elrond:
"But to the children of Elrond a choice was also appointed: to pass with him from the circles of the world; or if they remained to become mortal and die in Middle-earth."
Okay but why not to the children of
Elros?
It makes sense to me that
Arwen, Elladan and Elrohir should not be
automatically considered mortal, as if so, they would be
automatically sundered in fate from their parents. It also makes sense to me that they not be automatically considered immortal, given that they have some measure of mortal blood, and being denied the choice to leave the world and its time seems ungood for someone with even partial mortal heritage. In short it makes sense to me to extend the choice to them...
... now granted, given the choice of Elros (and his wife I guess), an automatic default to mortality would
not sunder children from parents. Okay so far, but they have a measure of "immortal" blood, and thus why are they essentially denied the choice to be judged as
Elves?
Tolkien himself
seems to answer your question at least a couple of times (if I recall correctly), in his letters. Someone will no doubt post these citations. I won't, because I'm lazy, because this post is already long, and because neither statement works fully for me in any case. Yep, in general, I don't always agree with Tolkien himself.
I have started arguments in empty rooms. LOL!
As an interesting side note, Tolkien's version of this section of
Quenta Silmarillion reads a bit differently from the 1977 Silmarillion. Tolkien wrote:
"Then Manwe gave judgment (...). Now all those who have the blood of mortal Men, in whatever part, great or small, are mortal, unless other doom be granted to them; but in this matter the power of doom is given me. This is my decree: to Earendil and to Elwing and to their sons shall be given leave to choose freely under which kindred they shall be judged."
We know Christopher Tolkien chose not to include this line, for whatever reason, in his Silmarillion published in 1977. And Tolkien never really fully updated the end of QS, so it's hard to say if he would have kept this idea moving forward (JRRT made some cursory corrections to the end of QS, but even Chistopher Tolkien warns that these shouldn't necessarily be taken as a stamp of acceptance regarding everything).
But if Tolkien carried this statement into a theoretical update, I'd note that according to this conception the children of Elrond would be automatically mortal-fated, and thus automatically sundered from their parents, if no "other doom" was granted to them...
... which ("other doom") was granted to them anyway 
And also, the children of Elros would be automatically mortal, if again, no other doom was granted to them. As I say, at least no automatic parental sundering here...
... but still 
