Why is the Free Rider Problem a problem? Why is it bad that someone can use something like a park or an emergency room without paying?
According to the standard criterion used to evaluate welfare (Pareto efficiency), the problem with 'free riding' is that goods are not produced even though they cost less to produce than they are worth to consumers. That means that we are forfeiting an opportunity to help people without harming anyone else.

To illustrate using a standard example, suppose that 10 people live on a street, and that each would value an additional street light at £100 (i.e. each would pay up to £100 to bring a new street light into existence). The street light would cost £101 to produce, but if produced can be used by all for free: this is known as 'non-excludability'. Will it be produced? If people are self-interested, then the answer is no. Although any resident would benefit from the street light, building it would cost them £101, which exceeds the benefit (£100). That's a problem since the total benefit (10 x £100 = £1000) greatly exceeds the cost of the light. If the inhabitants could be compelled to pay for the street light, sharing the cost equally, then they would all be better off.

I should stress that, from the perspective of Pareto efficiency, the problem with 'free riding' has nothing to do with 'equity'. To use your example of parks, the problem is not that people get to use the park without paying. Rather, the problem is that, since people are expected to free ride, the park won't be created in the first place.