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www.gov.uk/order-copy-birth-death-marriage-certificate
If you're in England/Wales then you can use this service to order up a full copy of a birth certificate which may give you more information such as parents' names or place of birth.
The other idea which immediately springs to mind is a DNA test through one of the sites like Ancestry. If your dad has siblings who would be prepared to test too, or even more distant relatives like cousins, then that might help. You can look up the percentage of expected shared DNA and then compare results - for example you could expect a sibling to share 50% of your DNA. If your dad tested, and so did a brother/sister, and the results were way lower than that. you know something's not quite right. It may also be that your Dad matches with someone else who has tested who he doesn't know, which could give avenues to explore.
If your Dad does suspect he was adopted, then he could request a copy of adoption records. This is quite a lengthy process, which involves counselling.
www.gov.uk/adoption-records
Of course if he was "adopted" but not formally, or was the product of an affair or something, that might not be recorded.
I'm sure you're aware but this could open up a whole can of worms for your Dad in terms of dragging up stuff from his childhood or questioning his entire identity. He'll need a lot of support, and should think long and hard about whether this is something he really wants to get to the bottom of, or if on balance, he'd be happier not knowing.