Blizzard has officially stated that the issue with not being able to complete the Dalaran Children's Week quests
this year, if you did them last year, will not be resolved before the holiday is over. They are not saying this is how it is supposed to work, but that they cannot fix it in time.
Bornakk -- Dalaran Children's Week Quest Line Issue
We are aware of the issue that is currently affecting the Children's Week quest line in Dalaran which does not allow players who have completed the quest line last year to take part in it again this year. We do not currently have a fix planned to address this issue at this point in time.
Bashlok -- Dalaran Children's Week Quest Line Issue
To clarify, this issue won't be able to be resolved for Children's Week this year. If you completed the quest previously you won't be able to complete it again for the other pet this year.