Verifying requirements before deleting a user-parent-student-school relationship
One of the biggest things you can do to clean up you code is to simply remove the unnecessary if-else blocks.  In many cases you have a return inside the first conditional, which means there is no reason for an else block at all.

For example, you code rewrite like this:

Student.findOne({parent_ids:parent_id},function(err,parentF){
    if(err){
        console.log(err);
        res.json({status:"error",message:"an error occurred"});
        return
    }
    if(parentF){
        res.json({status:"error", message:"you can not delete a parent who has students associated with it"});
        return;
    }
    Parent.findOne({_id:parent_id},function(err,parent){
        if(err){
            console.log(err);
            res.json({status:"error",message:"an error occurred"});
            return;
        }
        if(parent.school_id.length&gt;1){
            var a = parent.school_id.indexOf(school_id);
            parent.school_id.pop(a);
            parent.save(function(err,parent){
                if(err){
                    console.log(err);
                    res.json({status:"error",message:"an error occurred"});
                    return;
                }
                User.findOne({refid:parent_id},function(err,user){
                    if(err){
                        console.log(err);
                        res.json({status:"error",message:"an error occurred"});
                        return;
                    }
                    user.school_id.pop(a);
                    user.save(function(err) {
                        if(err){
                            console.log(err);
                            res.json({status:"error",message:"an error occurred"});
                            return;
                        }
                        res.json({status: "success", message: "parent removed"});
                        return;
                   });
               });
           });
        } else {
            Parent.remove({_id: parent_id}, function (err) {
                if (err) {
                    res.json({status: "error", message: err.message});
                } else {
                    User.remove({refid:parent_id},function(err){
                        if (err) {
                            res.json({status: "error", message: "parent user wasn't deleted"});
                            return;
                        }
                        res.json({status: "success", message: "parent data successfully deleted"});
                        return;
                    });
                }
            });
        }
    });
});


Hmm... half my post went away :(

I had also mentioned that you should look into naming your functions as a way to get away from the nesting that can be proliferated by anonymous functions.  You could add named functions for handling you error and success logging/response formation.   You could also add named functions to perform the callbacks, so as to flatten these callbacks out as well as provide re-use.

Finally, I think you have some style problems:


inconsistent use of semicolons at end of line
inconsistent spacing around flow control structures
inconsistent spacing around commas in function arguments and object literals
inconsistent use of camelCase vs. snake_case (you should probably just stick to camel casing in JS)


You should consider defining a style and enforcing it.