Recommendations for algorithm
A correlation matrix would help discover any pair-wise correlation (between any 2 variables). I think that would be a great place to start, as it is just one line of code for initial analysis.

After that, if you want to investigate "an increase in parameter 1 and parameter 2 is correlated with a fall in parameter 10", you could write code to form interaction variables like sum / product / divide and build a correlation matrix between the interaction variables vs. the original variables.

Personally, I found the corrplot R package useful in visualizing the correlation matrix. Hope this helps.