Problem 1: The students in my class (and their parents) are obsessed with marks. This means they get very nervous about exams.
Solution 1: Doing well can be motivating for learners, but in general, competition is not constructive in a class. Show learners how much you value their performance in class as well as homework and project work. With older children, you can give them some informal tests, and ask them to correct each other's work and to help each other to complete or rectify incorrect answers. Show learners that you value the progress they make, their teamwork and enthusiasm to learn. Give them plenty of feedback on this.
Problem 2: The parents think that traditional grammar and vocabulary tests are more important than continuous assessment, project work and performance in class.
Solution 2: Yes, this can happen. This is because classes now are probably very different from the classes their parents had. Try to explain to parents how their children are assessed, what the criteria are and the logic behind your approach to testing. Tell them about the advantages of your approach and, above all, the advantages for their children.
Problem 3: Parents believe that the most important role of the classes is to help students pass external exams.
Solution 3: It is possible to give learners lots of practice at question types from external exams. Obviously this is important, especially for parents. However, you could explain to parents that, in the long term, their children are going to need to speak English. This is the real reason that they are studying. A result in an exam is a very short-term goal.
Problem 4: Some students are often ill and can't do components of their continuous assessment.
Solution 4: In this situation I encourage the students to do as much as possible of the components that they have missed. Of course, you have to allow extra time to do this. Sometimes they can do this during the class, for homework, or during holiday time. At the same time, give your students a clear message that you are interested in the process, not just the result, and that the more work they do to catch up, the better it is for them.