How are go-arounds safe with close parallel runways?
The first thing I would point out is that LAX isn't going to be running simultaneous approaches on 24L and 24R or 25L and 25R. They'll select one of each of the parallel sets for the landing runways (typically the outer runways), and one of each for departing runways (typically the inner runways). For example, they might be landing on 25L and 24R and departing 25R and 24L. So, there's quite a bit of distance between those runways, and very little risk in a go around situation.

Here's a diagram illustrating LAX's typical arrival/departure runway usage to make it more visually apparent what I mean:



The second part of your question about an aircraft taking off at the same time as a go-around certainly happens, and when it does, ATC needs to take action to ensure safety and separation. ATC separation requirements can be satisfied by altitude, distance, or divergence (and, in some cases, visual contact). In the situation you described, divergence is the quickest and safest way to achieve separation. I believe the requirement is 30°, so tower would typically order a go-around and order a new heading at least 30° off of runway heading. The geography of the area, layout of the airspace, and the approach patterns would likely dictate the exact procedure used at each airport since their two primary goals, in order, are going to be 1. safety, and 2. getting them back in the approach sequence.

In the LAX example, a go-around on 24R would most likely get a right turn, and 25L would most likely get  left turn (away from the airport) so as not to conflict with other departing or arriving traffic. This is one of a few reasons the outer runways are preferred for landing.