I think it depends on the circumstances in which you are doing these additional hours. If they are simply being processed as overtime, you are going to come to a time soon where you might need to stop the overtime to be with your dad and the employer cannot take action against you if you are fulfilling your contracted hours. On the other hand, if they have given you a temporary contract for the additional hours then you may be obliged to fulfil ... unless there was an opt-out or review built in.
As far as the meetings go, it is quite common for employers have an attendance management policy which includes a manager review meeting at the third absence, regardless of whether those absences are one day or 3 months each. I imagine the policy then says that there are three stages to the attendance process, stage 1 being dealt with by the line manager, stage 2 & 3 being more serious and therefore dealt with by senior managers. Usually stage 2 involves a referral to occupational health or requesting a report from the GP.
They can only dismiss you if they can prove gross misconduct e.g. you worked for someone else while on sick leave. They can discipline you (such as a written warning or final written warning) if you have done something wrong which does not amount to gross misconduct, for example being off sick for more than 5 days and failing to produce a sick note. Many people supporting a relative at the final stages of their life get a sick note from the GP for stress, which protects them from disciplinary action. If you are concerned you can look up your rights in the ACAS handbook here http://www.acas.org.uk/index.aspx?articleid=2179 You could also phone ACAS on 0300 123 1100 for free advice
It is great to have a job that you love, but don't put yourself in a position that you might regret later - your dad will not be around for ever and it is a heavy heart that looks back and says "I wish I had had more time"