For intuition - imagine you are walking along a road in a forest and find 5 stones arranged in a line and ordered from smallest to highest. You are surprised to find such a constellation and start wondering: how likely is it for this order to occur without human intervention?
Here you can check various different "null" causes. One might be a hypothesis that wind placed the rocks in such a way. You want to obtain a p-value for this "null hypothesis" so you start placing various rocks on a road, wait for heavy winds, and check if the wind produced such a pattern. You repeat this experiment thousands of times and get a p-value: number of cases where wind, in your experiment, ordered the stones just as perfectly (or more) as you found them on the road. But you can also test for other possibilities. Like maybe that rain helped to order the stones. Here you repeat a similar experiment, but this time you wait for rain. Measure the numbers, get the p-value.
Choosing the right statistical tests boils down to choosing the null "cause" for your observation. In this example different tests would be: testing for rain and testing for wind.
Another detail is the order of stones. If you only want to check how often the wind can order stones from smallest to largest - you do a one tailed test. But you might also count reverse orders, where the stones get placed in decreasing order from largest to smallest.
So, as you see, in effect different tests answer different questions. When you choose one test - you ask one question, and if you choose another test you are asking a different question.
Now let's move on from the stones and briefly touch on parametric vs non-parametric. Say you measure the difference in height between girls and boys in a class. And you want to test the probability for the observed mean difference to occur in a case where there are no difference between those groups in the population.
But you have a problem here. There can be various different scenarios hiding under the "there is no difference in a population" statement, just like there were various alternative under "stones were ordered without human intervention". You have to choose more concretely.
One possibility is to do a non-parametric tests. Let's pick permutations. With the permutation test you shuffle the heights between the groups and measure the mean difference between boys and girls using these shuffled values. You repeat the shuffling thousands of times and deduce a p-value, as the number of shuffles that had higher difference compared to the one that was observed. But behind such a procedure - you chose a concrete null population, without, maybe, being aware. What you in effect stated is that your "null" population is only composed of the heights you observed in a sample. In other words, you made an assumption that your sample reflects the population perfectly. And this is one possible statistical test.
Another possibility is a parametric test. Let's forget the t-test for now and pick something more naive - you can state that each of those groups are drawn from the same normal distribution. If they are, the distribution of their sample differences should be centered around 0 and have a variance that is also deducible from your data. So you pick mean at 0, calculate the standard deviation for both groups, pool them, and now you have a parametric distribution to draw from. For the sake of simplicity, imagine you draw thousands of numbers from this distribution (thousand of mean differences under null of no difference) and deduce a p-value by counting how many of those drawn numbers exceeded the observed value. What did you do here? Instead of claiming that your sample reflects the population perfectly, you made a different assumption about the population - that it is normally distributed with mean 0 and some unknown variance. Then you computed this variance on your sample and used a "plug-in" rule to state that the population has exactly the same variance as the sample. This is a different way to obtain the "null population", and hence a different statistical test.
To summarise: under the hood all of statistical tests work in a similar way. Beneath each of them there is a "null population" lurking. And by selecting between different tests you select a different "null population". Which in itself leads to asking a different question about your observed data.
