Why do several (if not all) parametric hypothesis tests assume random sampling?

If you are not making any inference for a wider group than your actual sample, then there is no application of statistical tests in the first place, and the question of "bias" does not arise. In this case you would just calculate descriptive statistics of your sample, which are known. Similarly, there is no question of model "validity" in this case - you are just observing variables and recording their values, and descriptions of aspects of those values.

Once you decide to go beyond your sample, to make inferences about some larger group, then you will need statistics and you will need to consider issues like sampling bias, etc. In this application, random sampling becomes a useful property to assist in getting reliable inferences of the wider group of interest. If you don't have random sampling (and you don't know the probabilities of your samples based on the population) then it becomes hard/impossible to make reliable inferences about the population.

In real scientific research, it is quite rare to have data that came from true random sampling. The data are almost always convenience samples. This primarily affects what population you can generalize to. That said, even if they were a convenience sample, they did come from somewhere, you just need to be clear about where and the limitations that implies. If you really believe your data aren't representative of anything, then your study is not going to be worthwhile on any level, but that probably isn't true¹. Thus, it is often reasonable to consider your samples as drawn from somewhere and to use these standard tests, at least in a hedged or qualified sense.

There is a different philosophy of testing, however, that argues we should move away from those assumptions and the tests that rely on them. Tukey was an advocate of this. Instead, most experimental research is considered (internally) valid because the study units (e.g., patients) were randomly assigned to the arms. Given this, you can use permutation tests, that mostly only assume the randomization was done correctly. The counterargument to worrying too much about this is that permutation tests will typically show the same thing as the corresponding classical tests, and are more work to perform. So again, standard tests may be acceptable.

_{1. For more along these lines, it may help to read my answer here: Identifying the population and samples in a study.}

Tests like Z, t, and several others are based on known sampling distributions of the relevant statistics. Those sampling distributions, as generally used, are defined for the statistic calculated from a random sample.

It may sometimes be possible to devise a relevant sampling distribution for non-random sampling, but in general it is probably not possible.

These tests don't just incidentally assume random sampling. Instead, random sampling is what these tests are designed for.

Even if you've used careful study design to prevent all the other possible problems with a study (unbiased sampling, no measurement error, etc.), you still have to check whether sampling variability could be a problem. If you ran the study again, you'd presumably have gotten different participants, and thus different data, with different summary statistics. So, if the sampling variability is too large, then for instance you might get a positive treatment effect $\bar{x}_{meds}-\bar{x}_{placebo}$ this time but a negative effect next time, and thus you can't really trust the direction of the effect from any one study of this size.

Now, if you have a very large sample size, you'd probably get very similar summary statistics if you ran the study again. Is the sampling variability small enough that you can trust your statistics from this one study you actually ran? This question is essentially the only thing the classic hypothesis tests are checking for.

To do this check, hypothesis tests compare your actual observed statistic against a null distribution of "other statistics you could have seen if the null hypothesis were true." If you sampled your data randomly (simple random sampling / SRS, or independent & identically distributed / iid), this null distribution is straightforward to derive mathematically (or simulate computationally). But if your data are a convenience sample, then there's no clear way to derive/simulate an appropriate null distribution. If your actual statistic came from a convenience sample, it may or may not look "extreme" compared to a null distribution that assumes iid data... but that tells us nothing about how "extreme" it would look compared to other convenience samples.

Asides:

1. This is also why classic hypothesis tests get more complicated when you allow for "blocking" & other study-design features that minimize variance. If you know the data weren't sampled iid, there's no point in comparing your statistics against a baseline null distribution that was constructed under the iid assumption. Instead, you need to derive a null distribution that matches how your data were actually collected.0

2. As gung's answer points out, permutation tests can be appropriate if you carried out random assignment instead of random sampling. If that's true, and if it's a situation where the classic Z/t/etc tests are a good approximation for permutation tests, then you can use them. But you're using them because they approximate the permutation test, not because there was random sampling.