Learn English – Should I use “multiple” and “at once” in one sentence

Yes, but it's not really good English.

A participial phrasal noun ("picking up your litter") has been used as the subject of the main verb ("puts"). That's fine, grammatically.

The problem is that the phrasal noun is ambiguous, because it does not specify who would be doing the picking up. This is particularly bad for cases like this, where people are driving past the sign, and should not be taking the time to work out the exact meaning.

To remove the ambiguity, you can:

• Name the actor: You picking up your litter / Road workers picking up your litter / The queen of Sheba picking up your litter. These forms are awkward.
• Remove the genitive: Picking up litter puts... . This form removes the It's your fault! implication.
• Scrapping the whole sentence and starting again, but this time, saying precisely what you mean, using clear, unambiguous phrasing; e.g.: If you drop litter, people risk their lives picking it up.

"Rather than" can be followed by a gerund or a bare infinitive. (There are exceptions when it the two verbs it connects follow another verb that is the main verb of the sentence, but I'll ignore that case for this answer).

To take an example first of something that can't be used with such non-parallel forms:

*People might embrace their fetish or attempting treatment.

People might embrace their fetish or attempt treatment.

People must choose between embracing their fetish or attempting treatment.

Here or does require parallelism between verb tenses, so the first above is incorrect, while the other two are allowed.

Another informative incorrect case is to actually be parallel with a to-infinitive form:

*I would prefer to embrace my fetish rather than to attempt treatment.

I would prefer to embrace my fetish rather than attempt treatment.

Here we have to actually avoid parallel forms.

To come back to your question. In this case we can use both the gerund and the bare infinitive.

In all, either the form you found, or the form you suggest can both be used.

Edit:

It occurs to me that the distinction is hidden by the simple present (when not in third-person singular) and the bare infinitive look the same in English. It could be clearer in the past tense. First a case that does need parallelism:

He embraced his fetish or attempted treatment.

Here we posit two possible actions, that may or may not have taken place at some point in the past. We can even move the subject into what is parallel:

He embraced his fetish or he attempted treatment.

However, it would be wrong to have the same parallelism with rather than:

*He embraced his fetish rather than attempted treatment.

We are not saying that an attempt of treatment was something that happened in the past, or even that it is something that didn't happen in the past (though that is obviously implied), rather we are giving the possibility as one that hypothetically existed when he "embraced his fetish". It isn't something that exists in a moment of time, but adverbally modifies what is saying about the act of embrace, and no more has a tense than would an adverb like gladly, etc.

He embraced his fetish rather than attempt treatment.

He embraced his fetish rather than attempting treatment.