朴 as a surname in China somehow has three different pronunciations: piáo, pǔ, fú, at least according to this website. However, the 朴 surname in Korea is completely unrelated. The first person the 朴 surname in Korea was the founder of Silla, whose surname was apparently chosen to match the Korean word for gourd (which is also bak). Of course, hangul had not been invented yet, so the original Korean surname, Bak, had to be transcribed into Chinese characters. So, they chose 朴, which, at the time would have sounded quite similar to bak, at least looking at some of the reconstructions listed on Wiktionary for the first pronunciation, pò.
Now, this is my theory for the piáo pronunciation. When ancient Chinese people read texts about the Korean surname 朴, they obviously did not know about the actual Korean pronunciation, so they just defaulted to the standard pronunciation for 朴 as a surname, piáo, as piáo was the most common pronunciation out of the three possible 朴 pronunciations. This pronunciation carried over into modern times.
As for the pú pronunciation in Taiwan, pú was used in southern China, while piáo was used in northern China, according to this blog post. When South Korean President Park Chung-hee came to Taiwan to establish relations in 1966, the Taiwanese president told everyone to pronounce the Park Korean surname as pú because it is closer to the Korean pronunciation.