Yes, it is lawful for a parish council to consist totally of coopted councillors, although this is certainly not ideal.
Your remedies depend entirely on the willingness of more people to become councillors. When a casual vacancy arises, a poll will be arranged if it is requested by 10 or more electors (acting either together or individually). Although the poll will not take place unless there are more candidates than vacancies. Unfortunately the parish will bear the cost of the poll in this situation.
Every four years there will be an election for the entire council - although, again, whether there will be a poll depends on there being more candidates than seats. Putting up as a candidate is a very simple process and the officers of the election authority (usually the district council) are usually very helpful. The parish will not pay for this poll.