Over the years I have been asked this from time to time and it has always struck me as an odd question. My answer has always been, “Why the heck not”. Of course the parents of such a child are fearful that being at a funeral might somehow psychologically damage the child. That wont happen given that other circumstances are OK.
If at a funeral mother is really, really going to loose it emotionally then it might be a good idea to have someone else look after the child whilst there. There is nothing wrong, in fact it is psychologically good for a child to see mother cry at a funeral and even sob at a funeral if someone is nearby reassuring the child that mummy is OK. Which of course mummy is. Mummy will start sobbing and then mummy will stop sobbing and then life goes on. A child will not be psychologically damaged by viewing such an event.
This is quite different from the Child ego state feeling, knowing and believing the person is deceased. We all have a 4 year old inside us who will be there until the day we die. So we all have the ability to think and feel quite irrational things like a 4 year old can. Accepting that a loved one is deceased can be one of those occasions where the Child in us will think in a prelogical way and not accept the death in some way.
Funerals in this way can be very helpful. If the cultural rituals allow it I always suggest that there be a viewing of the body where the bereaved go and stand close to the body and touch it. The touching part in particular will let the 4 year old inside of us get some comprehension that the person is dead and gone. Often this can be quite powerful psychologically. And of course there is no reason why children cannot do the same.