Here is my theory:
#Theory#1
Assuming Yeon Soon Ho as ML's Father.
FL revealed that the man holding ML in the old picture was her father.
Pi Madame suggested that the same man(Yeo Soon Ho) holding ML could be his biological father. ML’s mother was standing nearby, the same woman from ML’s childhood memories. ML remembered the final moments with his mom and dad, traveling in a car on a rainy night.
Also, the picture was taken by FL’s mother (Deok Hui), as she admitted herself. At that time, II Do was already in the picture, which suggests a possible affair with FL’s mom. Out of jealousy, II Do could have plotted to kill FL’s real father to remove him from his love interest (ML's Mom). At the same time, FL's mom could have also plotted to kill her first husband's (Yeo Soon Ho) mistress (ML's mom).

In the recent episode (Episode 10), there was an intimate scene between the older couple, lying in bed and discussing whether they could kill Dong Ju. Their thoughts aligned disturbingly well, which strongly suggests that FL’s mom may have been involved in the plan to kill FL’s father. As she might have assumed that ML was her ex-husband's son (Yeo Soon Ho), that might be why she harbored ill intentions, as he could become a potential competitor in the battle of inheritance.
What hasn't been shown is whether FL was already born when the picture was taken. I don’t remember the ages of ML and FL—do you guys know?
This is one Theory, assuming Yeon Soon Ho as ML's Father.
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#Theory#2
#II Do as the father of ML
Deok Hui and Yeon Soon Ho were once married. They had Eunnam (FL). The exact timeline is not known, but we assume that it was after ML was born.
Deok-hui didn't like her husband getting close to ML's mom.
II Do didn't like Eunnam's father getting close to ML's mom.
So, he might have caused an accident that was meant to kill her husband, but he also accidentally killed his love interest.

ML got lost in the car accident and grew up as Seo Dong Ju.
The post-accident resulted in the loss of their loved ones, both Il Do and Deok Hui, and they got closer and they got married. They give birth to Heo Tae Yun (FL's half-brother).
So, ML shares the same half-brother, even though they don't have any blood relation.
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#Theory#3
#ML's Father is someone who started or was connected to the Dandelion Team
Dandelion team is a secret Service administered by the National Security Agency to curb corruption among influential individuals.
Hypothesis 1: ML's father could have discovered some secret that angered the villain, so he was killed.
Hypothesis 2: ML's father hid to escape from his perpetrators.
The identity of ML's father has not been revealed to us, but he could be someone they have already introduced to us (Assuming they follow Knox guidelines).
Could it be
Yoon Soon Ho?
Choi Gu Hyeon?
Yeon Chi Seon?
Nam sang cheol?
Or....
Could it be Yeom Jang Seon?

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#Theory#4
Alternative Theory:
If we assume the directors used Foreshadowing,
It could be either II Do or Yeon Jang Seon as the whole show revolves around them.

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Which Theory do you think is the most plausible?