The job of a parent is to protect, guide, nurture, and discipline if necessary, to ensure their child has the skills & mindset he or she needs to function well as an independent adult. To put it another way: A child has lots of friends of his or her own age; what he/she needs is a responsible adult.
Primarily a disciplinarian - but a good enough friend that the child can confide in the parent and discuss their problems.
Children should have many other friends, but will only have a few figures that can help them grow into responsible adults.
Definitely in-between. Not so strict that they hate you. If you let them make their own mistakes after warning them once, they'll learn better that way.
Both; your child needs to know what is good and what is bad. Being your child's best friend is great, but don't let them get away with being a rude brat.