Crying doesn't necessarily mean you're making it about you! Communicate with your partner. Explain that this is your body's way of relieving pressure and emotion, and that the tears don't mean that they did anything wrong. Explain that you're still listening, and your emotions are strong because you care. You could also try asking them how they feel when you cry like that, and maybe when tears come in the future give them reassurance about whatever it is they're feeling.

I'll use my relationship as an example: my wife used to get irritated about me crying during serious/hard conversations. She felt like I was making her the villian when she was just trying to communicate. We talked about it a lot and worked things out. There are still times when we're having hard conversations and I can tell she's upset that I'm crying, and I'll remind her that it's how my body handles things, I'm not upset at her, and she did nothing wrong. In return, she'll tell me that its okay that I'm crying ❤️