Linear Regression. Interpret weights as an effect on a prediction?
In a basic linear regression, can I use the weights of each
  explanatory variable to describe their relative effects on the
  predicted value? If parameter A has a weight of 100, and parameter B
  has a weight of 10, can I say that parameter A has 10 times more
  effect on the outcome than parameter B?
  Blockquote


Somewhat yes. Precisely it means if A changes by one unit the outcome will change 100 times, on average (while B is frozen because it is a multiple linear regression -> you have more than one independent variable).

There are tons of examples and blogs explaining it, see for example here .
Also you can analyze your regression model more by doing some statistical tests like identifying how variables are correlated using Variance Inflation Factors and estimate the statistical significance (p-value, F-test), here is a very nice intro. 


  An example: From a weather data set, I used two parameters, humidity
  and pressure, to make a prediction on temperature.


I do not see any problem with your analysis. As you said you see an inverse correlation between humidity vs. temperature in your plot as well as via your -.32386 weight. While the weight of -.02219 for pressure is nearly zero and negligible and obviously you won't see such relationship between pressure-temperature.