Avia ante lararium parvum stat et paucis verbis deam salutat.

Because ante as a preposition takes the accusative case.

So:

• lararium = accusative singular of lararium
• parvum = accusative singular neuter of parvus, agreeing with lararium

Even though English says in front of the small shrine, Latin uses ante + accusative:

• ante lararium parvum = in front of the small shrine

This is true even when the meaning is static, not motion toward something.

Avia is nominative singular.

It is the subject of both verbs:

• stat = stands
• salutat = greets

So avia is the person doing the actions:

• Grandmother stands ... and greets ...

Because deam is the direct object of salutat.

• dea = goddess in the nominative
• deam = goddess in the accusative

Since the grandmother is greeting the goddess, the goddess receives the action:

• deam salutat = she greets the goddess

Paucis verbis is in the ablative plural.

• paucis = ablative plural of pauci (few)
• verbis = ablative plural of verbum (words)

This is an example of the ablative of means/instrument or a very similar idiomatic use:

• paucis verbis = with a few words / in a few words

Latin often uses the ablative where English uses with.

Because Latin idiomatically says with a few words, not with little speech in this sentence.

So:

• verbum = word
• verbis = with words / by words

The phrase paucis verbis is a common expression meaning:

• in a few words
• briefly

In Latin, adjective placement is more flexible than in English.

Both of these are possible in principle:

• parvum lararium
• lararium parvum

What matters most is agreement, not position.
Here parvum agrees with lararium in:

• gender: neuter
• number: singular
• case: accusative

So lararium parvum still clearly means small shrine.

Because the subject, avia, is singular.

Latin present-tense verb endings show the subject:

• stat = he/she/it stands
• salutat = he/she/it greets

The ending -t marks third person singular.

So the sentence has one singular subject doing both actions:

• Grandmother stands ... and greets ...

A learner would usually look them up like this:

• avia → avia, aviae = grandmother
• ante → ante = before, in front of
• lararium → lararium, lararii = household shrine
• parvum → from parvus, parva, parvum = small
• stat → from sto, stare = stand
• et → and
• paucis → from pauci, paucae, pauca = few
• verbis → from verbum, verbi = word
• deam → from dea, deae = goddess
• salutat → from saluto, salutare = greet

A lararium is a household shrine in a Roman home.

It was a place for honoring household gods, especially the Lares and often other deities too.
So in this sentence, the image is of the grandmother standing before a small domestic shrine and greeting a goddess there.

Classical Latin has no articles equivalent to English the or a/an.

So:

• avia can mean grandmother or the grandmother
• deam can mean a goddess or the goddess
• lararium parvum can mean a small shrine or the small shrine

You figure out which is meant from context.

It can mean both, depending on context.

• spatial: in front of
• temporal: before

In this sentence it is clearly spatial:

• ante lararium parvum stat = she stands in front of the small shrine

If it were temporal, the context would be about time rather than location.

The word order is natural, but Latin word order is more flexible than English.

This sentence is:

• Avia — subject first
• ante lararium parvum — place
• stat — verb
• et paucis verbis deam salutat — second action

A more literal breakdown is:

• Grandmother
• in front of the small shrine
• stands
• and with a few words
• greets the goddess

Latin relies heavily on case endings, so word order can vary for emphasis. But this version is straightforward and easy to read.

Because once the subject avia has been stated, Latin does not need to repeat it.

Both verbs are third person singular:

• stat
• salutat

So the sentence naturally means:

• Grandmother stands ... and greets ...

Repeating the subject would usually be unnecessary unless the writer wanted emphasis or contrast.

Yes, that is a good idiomatic translation.

Literally it means:

• with a few words

But in smoother English it can be:

• briefly
• in a few words

So paucis verbis deam salutat can be understood as:

• she greets the goddess briefly
• she greets the goddess with a few words

Both capture the sense well.