You have explained that you work part-time and that your employer does not pay you for your CIPD class, which takes place outside of your contracted hours. However, your full-time colleagues do receive payment for the time they attend their course. I presume that the main reason for this is that they attend university during their contracted working hours. You have asked whether this is fair.
Under the Part-Time Workers (Prevention of Less Favourable Treatment) Regulations 2000, an employer must not discriminate against part-time workers. You therefore have the right not to be treated by your employer less favourably than your employer treats a comparable full-time worker, in respect of the terms of your contract of employment, or by being subjected to any other detriment by any act, or deliberate failure to act, of your employer.
The Regulations therefore give part-time workers various rights, for example, the right to receive the same hourly rate as comparable full timers and not to be excluded from training, simply because they work part time. This right applies where the reason for the treatment (which in your case would be non payment for the time when you attend at university), is that you are a part-time worker and the treatment is not justified on objective grounds. I do not believe that your employer could justify this treatment.
You should speak to your manager and explain the situation and what is happening, noting that you are not being paid for the time when you are at university, whereas your full-time colleagues receive payment for the time when they are at university. You can explain that you have taken employment law advice from a solicitor and that you understand this is in breach of the Part-Time Workers Regulations and that you must be given the same rights as full-time workers. You should state that if your employer does not pay for the time when you are at university, this may mean that you cannot continue with your studies, due to the lack of assistance you are receiving. My advice is that you explain that you do not wish to fall out with your employer over this and would very much appreciate it if your employer could reconsider the situation. If your employer agrees to pay you for your time when you have been at university (and hopefully make a backdated payment to you in this respect), the matter will be resolved. If your employer does not agree to this, I would then advise you to lodge a formal grievance in writing, i.e. send a letter to your employer stating the above and asking them to re-consider.
If your employer does not subsequently agree to make the relevant payments to you, you may potentially be able to bring a claim under the Part-Time Workers Regulations for discrimination and/or bring a claim for constructive dismissal. However, I would not advise you to take this course of action without seeking further legal advice.
The above is only relevant if the course that you are studying for is relevant to your work, unless of course your employer pays for your full-time colleagues to attend university for courses which are not related to their work.