We've all heard the expression "failure to plan on your part doesn't constitute an emergency on my part". It's a good point. One parent may have a sense of urgency because they want the other parent to respond quickly, but in reality, the issue or question is not truly one that requires an immediate response. "Rapid response" texting or messaging is generally not a good idea in conflict situations. A good rule of thumb is to sleep on any response before sending it.
Your co-parenting relationship should be viewed as a business relationship; try to take the emotion out of it. Use business etiquette and rules as a guide for what is reasonable in the context of co-parenting communications. A 24-48 hour response time is usually not unreasonable. You may check back with the co-parent politely to confirm the message was received if there has been no response after 72 hours. If you are on the receiving end of an inquiry and for whatever reason, you are not ready to respond yet, after 24-48 hours, you should respond to the co-parent acknowledging receipt of the inquiry and letting the parent know when they might expect an answer (eg: "I'll check and get back to you by the end of the day tomorrow" or "I won't know my schedule until next week; I'll let you know Monday").
What is a true emergency? Blood, guts, and hospitals. Short of that, do not cry wolf and call something an "emergency" just because you want the other parent to respond quickly. Routine matters can wait a week or more before a decision is made; important matters may need to be addressed within the week; urgent matters should be addressed within 24-72 hours; a crisis warrants a response in 24 hours; and an emergency should be addressed asap.
Example of a routine matter: is your child going to sign up for a sporting team next season.
Example of an important matter: the teacher assigned a project due Friday.
Example of an urgent matter: the child is sick, but he/she is not going to the hospital or to see a doctor. No broken bones, no blood, no medication.
Example of a crisis: the child forgot his inhaler but he/she is not currently symptomatic or sick.
Example of an emergency: the child broke his/her arm during practice and is heading to the hospital.
See if you can get your co-parent to agree on time frames within you will each respond for any non-emergency matter, so you both have similar expectations.
Was there ever a time when you and your co-parent differed about the urgency of a situation and the appropriate response time? Describe the situation and explain your thought processes.