Learn Swedish: Du bör inte bekymra dig för din gamla pappa. - You should not worry about your old father.

In Swedish, many verbs that express an emotional shift or a state of mind are reflexive. The non-reflexive verb att bekymra means 'to trouble' or 'to cause worry to' someone else. By adding the reflexive pronoun dig, it literally means 'to trouble yourself,' which is how Swedish expresses 'to worry'.

Could I use the preposition om instead of för here since they can both mean 'about'?

No, the verb bekymra sig specifically pairs with the preposition för when stating what you are worrying about. You might be thinking of bry sig om (to care about), which does use om, but mixing them up is a common trap!

Why does the adjective gamla end in an '-a' when pappa is a singular en-word?

Because it follows the possessive pronoun din. In Swedish, possessive pronouns make the noun phrase definite, which forces the adjective into its definite form (usually ending in '-a'). Notice that the noun itself (pappa) stays in its indefinite form after a possessive.

Is bör the same as skulle when translating 'should'?