If an employer puts you into a training program, shouldn't that time be paid?

It depends on how the program is set up. If it is training "on the job," offered and conducted by the employer at the work site, it would be work; work is anything you do at the employer's or at a client's location, whether it is "productive" or not. Note that this would not prevent you from agreeing to do it on an unpaid basis, if that is the right choice for your career and relationship with the employer.

If this training is offered by some other provider (e.g. a class or training program) and if it represents not specific instruction on how to do the employer's work, but more general training or credentials that  someone in your position should have, then it is likely not work--it is the equivalent of continuing professional education or schooling, which an employer may require (and can even choose  to pay for), but for which the employer does not need to provide salary.