Mdthbs

I pass Salam a lot, almost to every man I pass by, but when I say Salam (not to my father) when my father is with me, he gets angry and shouts at me. He says that if they wanted Salam they would have said it, and that I should say it to people I know, and some other excuses.

Though his arguments are weak, but since he's my father, and kind treatment of parents is obligatory, should I stop saying Salam when he's around, since Salam is a (very recommended) Sunnah?

Initiating salam is Mustahabb while obeying father is Wajib, the latter takes precedence. Also remember the Hadith:

I asked the Prophet (ﷺ) "Which deed is the dearest to Allah?" He replied, "To offer the prayers at their early stated fixed times." I asked, "What is the next (in goodness)?" He replied, "To be good and dutiful to your parents" I again asked, "What is the next (in goodness)?" He replied, 'To participate in Jihad (religious fighting) in Allah's cause."

"Obedience is only in righteousness (الطاعة بالمعروف)" means you do not obey anyone who commands to commit misdeeds or to abstain from obligatory duties. However in affairs in which you have choice you can obey parents even if they stop you from a good thing.

'Abdullah ibn 'Amr said, "A man came to the Prophet, may Allah bless
him and grant him peace, and said, 'I have come to make you a pledge
that will do hijra although I have left my parents in tears." The
Prophet said, 'Go back to them and make them laugh as you made them
cry.'"

{https://sunnah.com/adab/1/19}

A man emigrated to the Messenger of Allah (ﷺ) from the Yemen. He asked
(him): Have you anyone (of your relatives) in the Yemen? He replied:
My parents. He asked: Did they permit you? He replied: No. He said: Go
back to them and ask for their permission. If they permit you, then
fight (in the path of Allah), otherwise be devoted to them.

{https://sunnah.com/abudawud/15/54}

You say that "Salam is a (very recommended) Sunnah", this implies that it is not obligatory. If this is your belief then my opinion is that you should take a lesson from the above hadith and keep your father happy, at the very least while he is observing you.

"A Muslim has six rights over another Muslim. When he meets a Muslim, he greets him; when a Muslim invites him, he accepts his invitation; when a Muslim sneezes and says, 'alhamdulillah (praise be to Allah)', he says, 'yarhamukallah (may Allah have mercy on you)'; when a Muslim becomes ill, he visits him; when a Muslim dies, he takes part in his funeral; he loves for his Muslim brother what he loves for himself."

(Darimi, Istidhan: 5; Ibn Majah, Janaiz: 43)

According to above it's an obligation to greet muslims with saying salaam too. As you say it's obligatory to obey parents and your situation may look like conflicts with that. If your father stops you just because he's annoyed with the repetition or some other "weak" reason you shouldn't stop saying it since obligation needs to be on righteous things.

I recommend you not do it if it is not welcome by the receivers or your father.

Unfortunately I have to say, one of the calamities that have befell the Islamic world is soulless adherence to the ritual without knowing their purpose and their underlying moral foundations. This happens when law is overrated against ethics and spirituality, (a shortcoming that is affecting most of us Muslims today, I believe!)

Saying salaam as a greeting in Islam is meant to be an opening message of peace, sympathy and kindness to your fellow humans, and as that it would have to come from a genuine attitude of peace and kindness in our hearts in which case it would be inevitably expressed in our mood, tone and bodily expression when uttering it.

So if the lively spirit is there, you can guess it would be very unlikely that someone would not welcome your greeting because all people under most circumstances would enjoy kind and cheerful expressions of friendship even from passing strangers, in which case your father wouldn't most likely object to it either.

But if greeting is interpreted by fellows as a show of piety, a value-laden imposition or a nuisance then you are basically sending the wrong message which is not the message of "salaam" which is about peace, sanity and good-faith.

Let me demonstrate this by a more cultural example. We in Iran know that in Islam we are supposed to say the Islamic greeting in full, i.e. "as salaam alaykum" which translates to "peace be upon you" instead of saying only "salaam" but sensitive Iranian Muslims may still intentionally drop "alaykum" because, at times, using it can be misinterpreted as a forceful show of piety than a heartfelt good wish for someone. So a good rule of thumb is that whether we can say the same Arabic thing in Persian without the pretentious pious connotations of saying it in Arabic and check to see whether the feeling of telling someone literally "Oh, Ali, peace be upon you today. How are you doing?" or "Oh, sir, peace be upon you. I have a question" sounds genuine to us or rather just redundant, slushy or artificial.

In fact we would rather try words that are not ideologically loaded at all to express a genuine friendly spirit like saying "Salaam Ali jan" (Hi, dear Ali) for a close friend, or "Salaam doost-e aziz" (literally, hello dear friend) for a warm stranger. Yes, we are not saying those specific Arabic words "salaam alaykum" but we are actually expressing the very spirit which Prophet Muhammad (pbuh) expected people when he recommend Arabs of his time to employ certain words to express greeting.

In other words, it was not the words per se that a supreme moral teacher like Muhammad (pbuh) was so concerned about but the spirit that they carried in the culture of his time.