The question is ultimately not the one of physics, but of statistics. It is for a good reason that particle physics remains one of few fields of physics where statistics is still practiced on an advanced level (in many other fields high precision of measurements reduced the need in statistical analysis to calculating standard deviations). In particular, the statistics chapter in PRD is an excellent crash course about statistical analysis.
How many parameters is many?
In physics we are used to models where the number of parameters can be counted on our fingers, because we are aiming at understanding the elementary interactions/processes/etc. Describing any real world phenomena necessarily results in combining many elements and using more parameters. The models used in engineering, e.g., to design airplanes, or in government planning contain hundreds or thousands of parameters. The high promise of machine learning is due to the modern computational ability to use models with millions of parameters, often having very obscure meaning (to humans) - but they still work very well, as we see by Facebook tagging photos or growing quality of Google translate.
How much data?
Whether we have too many parameters depends on how much data we have. The rule-of-thumb is having more data points than we have parameters. However, more principled approaches are built around the likelihood that is the probability of observing data, given our values of parameters: $$P(D|\theta).$$ Model in this context is the means of expressing this relationship between the parameters and the data mathematically.
Now, if our model is any good, the likelihood will be increasing as we increase the amount of the data (the number of the data points) - although this increase is not strictly monotonuous, due to the random effects. If this does not happen, our model is not good - perhaps it is too simplistic, has too few parameters - this is called underfitting.
Given a wealth of data, the model with more parameters will generally result in higher likelihood - this is where the problem raised in the OP lies. Let me note in passing that we can never prove or disprove a model by itself - rather we compare different models and choose a better one. A model with more parameters may be simply better, because it better aporoximates physical reality. But such a model can result in a higher likelihood simply because we have more parameters to tune - this is what we call overfitting.
Methods have been developed for correcting for the number of parameters when correcting model. One of the most well-known is Akaike information criterion (AIC), where on compares quantities $$AIC=k_M -\log P(D|M),$$
where $k_M$ is the number of parameters un model $M$. The model with the lowest value of AIC is then considered to be the one that achieves the best results with the smallest number of parameters.
Lest this simple criterion appears too intuitive, let me point out that justifying it rigorously requires quite a bit of math. There exist also more elaborate versions, as well as alternative criteria, such as Bayesian information criterion (where $k_M$ is replaced by its logarithm).
This is how choosing the best model is done in a nutshell. The physics comes in in formulating the logically motivated models to choose from. I suspect that, if we look at the publications in the times when the standard model was formulated, there were quite a few alternative proposals, and even more were probably floated in discussions among the scientists. Yet, the beauty of physics is that it allows to significantly narrow the choice of models - as alternative to machine learning approaches, where all possible models are equal, and the choice is based solely on their compatibility with the data.