There are two things to consider here.
First, the p value reported for an individual coefficient in a standard linear regression is based on a t-test. As that test uses an estimate of the standard error taken from the data, the reliability of the standard error estimate depends on both the number of observations and the number of parameters that are estimated. The degrees of freedom in the t-test decrease as the number of estimated parameters increases: $(N-P)$ for $P$ parameters estimated from $N$ observations. That's the level of correction for numbers of parameters in the p-values reported by standard statistical software from the t-tests for regression coefficients. Note that if the number of parameters estimated is much smaller than the number of observations, this adjustment won't make much of a difference.
Second, which I take to be the basis of your question, is the matter of multiple comparisons, the idea that the more significance tests you perform the more likely you are to find one nominally "significant" result simply by chance: a test at p < 0.05 when a null hypothesis is true will be a false positive in 1 out of 20 comparisons. Standard statistical software and published tables of estimated regression coefficients typically do not correct coefficient p -values for multiple comparisons. See the discusssion on this page for some thoughts.
One exception is software used for big-data analyses like gene-expression studies, which builds in important corrections for multiple comparisons. If you tried to rule out any false-positive results among the thousands of nominally "significant" coefficients you would necessarily miss a lot of true positives. In such situations the false discovery rate, the fraction of "significant" findings likely to be false, is typically controlled. See this page among many others on this site.