In this context, 'volunteer some time' means that the parent will give some hours from their day for assistance. What they do, is up to whomever would supervise the work. Perhaps the teacher will get the parents to supervise play, or organize activities, or do light cleaning / tidying. Technically, to volunteer is as follows: A person requests one helper, from a group of many potential helpers. An announcement is made to all - 'a Volunteer is required for a task'. All the potential volunteers will consider the request, until one person says they will perform the task They are now 'the volunteer (noun)'. The person is said to have 'volunteered' (verb). Another, more abstract use of the verb is 'to volunteer someone'. In an army scenario, it is typical that the commanding officer will select his volunteer. This is a little nonsensical, but the army is like that. the CO will say 'you just volunteered' to the selected soldier. So, this verb is 'to volunteer somebody' is to nominate another person for a task where really the only option given was to volunteer yourself, or say nothing at all. In the school context, it is usually a request for a persons time, that is made to all the children's parents. Perhaps one parent can do 2 hours per week, another can do 8. So each parent volunteers as many 'hours' as they can. Because the parents control their own time, it is applying the verb to their time, a little like the army example, but without the nonsensical meaning.

Usually it means they are scheduled to go in and do something to help out in the school. I'm not sure which culture you are in, but in America, it would never be cleaning, as far as I know. Parents often help with administrative tasks, or cafeteria or playground supervision, etc.