I would just ignore the snark and rise above it. It will frustrate her when she can no longer get a rise out of you because you don't care any longer, and then she'll stop.
Regarding the co-parenting its a good idea to discuss in detail what it means to you. For instance, what scenarios can each of you address individually, and what needs to be addressed together?
Who is responsible for scheduling doctors' visits and which parent needs to do the driving?
When a child is sick at school, is one person primarily responsible to leave work and pick them up, or is it based on whoever responds soonest?
On a snow day when school is cancelled, who is responsible for doing the driving between houses for the parenting handoff and what time does it occur? When school would have started, or when school would have ended?
(etc. etc.) The more of this you can spell out and put on your website the better, because it will preempt conflicts as each case arises.
FWIW, I don't think it's practical for you two to be on a conference call for every teacher conversation, the logistics are too difficult. Generally it works well for one parent to have an initial conversation and if it seems like a major issue, ask for a follow-on with both of you. If it's a minor issue just deal with it. That obviously requires you two to make judgement calls, so in that case you should discuss what would fall into the categories of major and minor.
Finally, you don't care if she looks like an absentee parent, that is not your concern.
Acc
Married 18, Together 20, Now Divorced M: 48, W: 50, D: 18, S: 16, D: 12 Bomb Dropped (EA, D): 7/13/11 Start Reconcile: 8/15/11 Bomb Dropped (EA, D): 5/1/2014 (Divorced) In a New Relationship: 3/2015