There are lots of ways you can be visible. At my school patents don't go to the school to drop off their children (it's a special school so most children get transport), so I go to the parents coffee morning that's held twice per month.
I also go to the new parents' information morning and have a 'slot' where I introduce myself and explain the role of governors, what we do and (crucially) what we don't do. At that meeting I always say 'I'm often in the school, so if you see me and you need anything, feel free to ask'.
We always send out a parent questionnaire each year, to find out parental views.
As governors we are one of the 'faces' of the school. So it's important to be open to parent interaction. Its just vitally important that we don't slip into being a 'sounding board' for parents who have gripes about teachers, lunch box policies, absence policies, the HT's personality, etc.
We need to stay strictly strategic in our involvement in the school. We're there to ensure: that the policies and procedures of the school are fit for purpose and that the evidence we see of their implementation demonstrates that they are being effectively applied; that there is evidence of good progress in pupils (in a broad sense -we should never be aware of the grades of a named child) and that where progress is less than good, measures are being taken to support those children; that financial policy is robust, finances are in order and there is not a huge over spend nor a huge under spend -the funds are there to be spent on the children and huge under spends tend to be clawed back by the LA; that personnel procedures are rigorous and applied well, seeing evidence of the quality of teaching (again, broad strokes -you should never know which teacher has achieved what standard), as well as measures taken when teaching is below standard; that pupil premium funding is being maximised and used to benefit the qualifying cohort, with evidence of the impact it is having.
So, you can see, being a Governor should never be about discussing that Mrs Matthews is upset that her child can't have two weeks off to go to Cyprus. That's for the HT to deal with.