There are different ways to say father and mother in Chinese.
The most popular one is 爸爸 (bà ba) and 妈妈 (mā ma), or just 爸 (bà) and 妈 (mā), which can be combined into 爸妈(bà mā), meaning parents.
A bit more formal way to address father and mother is 父亲 (fù qīn) and 母亲 (mǔ qīn), which can be combined into 父母 (fù mǔ) or 父母亲 (fù mǔ qīn), also meaning parents.
In Hong Kong, I quite often hear children address their father as “Daddy” and mother “Mommy”. It is not that Hong Kong children speak English with their parents. They use Cantonese when speaking to their parents, but addressing them in a different language.
Another way to address father and mother is 爹 (diē) and 娘 (niáng). And the word for parents becomes 爹娘 (diē niáng). I read these terms many times in novels, heard these terms many times in films. But I personally don’t know anyone who addresses parents this way. Some say that these terms are more commonly used in rural areas. I can not verify that.
There is another way to address father and mother in Chinese, 令尊 (lìng zūn) and 令堂 (lìng táng), which strikes me as very old fashioned and extremely polite and respectful. I have only seen these terms appear mostly in classic works. And I don’t think these two words can be shortened into a two-character word for parents. On the contrary, it is often expanded into 令尊大人 (lìng zūn dà rén) and 令堂大人 (lìng táng dà rén), to be even more polite and respectful.
In Hong Kong, when you go hiking, you might spot tombs along the trails. If you read the tomb stones carefully, you may learn two new words for father and mother in Chinese, 考 (kǎo) for a deceased father and 妣 (bǐ) for a deceased mother. Deceased parents would be 考妣 (kǎo bǐ).
Such as the following photo shows (read from right to left as in traditional Chinese books):
In Chinese traditional culture, father always associates with being strict and mother always associates with being benevolent, hence the concept of 严父 (yán fù) and 慈母 (cí mǔ) is deeply embedded in Chinese literature. Father is to be feared and mother is to be loved.
In reality, mother can also be very strict. The term “tiger mom”, 虎妈 (hǔ mā), has been very popular to describe Chinese moms. Although this is a relatively recent term, a strict mother is not a new concept. There are many stories about strict mothers in China. One such famous mother is the mother of Mencius (c. 371–c. 289 bc). One story goes that she destroyed perfectly good cloth just to teach her naughty son a lesson.