Like Laser28, I would like to know the location. That makes a difference. There are several possibilities to cause your description of distress. These include; alkali-aggregate reactivity, chloride intrusion, freeze-thaw deterioration, and wet/dry cycling.
The skewed joints and parallelogram design sounds like an experiment that happens about every 20 years or so, but it never seems to catch on mainstream (no pun intended!). The regularity of joints in rigid pavements and bridges causes a "sing-song" pattern of vehicle noise that's annoying to drivers. Every now and then someone in a DOT will decide to skew and re-space the joint pattern to break up the noise pattern. It works to some degree but is more costly to implement.
Back to your observations....stalactites from cementitious materials can occur with or without carbonate aggregates. It is a sign of general deterioration of the concrete from wetting/drying cycling. It is comparable to efflorescence on brick from calcium leaching from the mortar. The same happens in this case. Cement's largest constituent is calcium, in one form or another. Calcium easily goes into solution in water, is carried to the surface where it reacts with carbon dioxide, and calcium carbonate is formed. It progressively thickens into the formations you see.
What to do? As JAE suggested, some testing is warranted for chloride intrusion. There are generally two ways to assess chlorides in concrete. One is using a chloride permeability test, commonly used in Canada. It provides you with the potential for chloride intrusion in THAT concrete. The other is to test for soluble chlorides in the concrete from cores or from drill cuttings. This can be done in a gradient fashion to show concentration with depth, but you have to be careful about cross-contamination of the gradient samples. I wrote a procedure to do this about 15 years ago. Post your email and I'll send it to you.
A petrographic examination of a few cores would be a good idea to check the integrity of the concrete. Considering its age, it might not have been air-entrained, so this would be good to know. Further, the integrity of the paste-aggregate bond can be determined from petrography, as well as indications of alkali-aggregate reactivity.
Finally, a load test of the bridge for its rating should be considered. This should be a dynamic load test with real-time monitoring of load and strains to check for the bridge's response to load. With strategic placement of linear deflection devices and strain gages, a good idea can be gained of the load response and that can be compared to a model of the structure to check theoretical vs. actual. This helps you assess the condition of the bridge. Keep in mind that you will likely have to do some "guessing" (educated of course!) as to the original reinforcement or some significant nondestructive mapping to create your model.
Good luck!
