The Governing body should only be involved in complaints if the complaint has not been resolved at school level. Most complaints should be able to be resolved by the class teacher. Some may need to be resolved by Key Stage leader, or Deputy Head. Some may need the Head Teacher to resolve them.
I entirely agree. But how the Governing Body is involved is set put in policy and varies from school to school.
Some Schools agree a policy in which the Chair (or a named Governor) is made aware at an early stage, in the knowledge that they would be excluded from any subsequent complaints panel. Other schools ensure that all Governors remain untainted until a complaints panel is convened.
While a Formal Complaints Panel is the point of last resort, it is clear from the comments on this thread and others that many parents want Governor "involvement" at a much earlier stage in the process. Involving an appropriately trained Governor to support conflict resolution at the informal stage can often avoid escalation.
While it's not clear how the telephone conversations that the OP has had with the Chair have come about, I have supported CoG in similar situations who have liaised with the parent on the phone or met face to face, with the full knowledge of the HT, in order to secure a resolution and avoid a "formal" complaint. Of course, that is only possible if the School Complaints Policy allows it.
Some of the comments about the CoG on this thread have been critical, when in fact, they may have acted entirely appropriately and in line with school policy. It's just not possible to judge without more details.