По выходным мой дедушка возит внука в парк.

This is a very common Russian time expression.

по + dative plural can mean on / during / every in a repeated-time sense. So:

• по выходным = on weekends / at weekends
• по утрам = in the mornings
• по вечерам = in the evenings

It usually suggests a habitual or repeated action, not just one specific weekend.

Because after по in this kind of time expression, Russian uses the dative plural.

The full expression is historically/grammatically connected to something like:

• по выходным дням = on weekend days

Very often дням is omitted, and выходным remains by itself.

So:

• выходные = nominative plural, the basic dictionary form for weekends / days off
• по выходным = dative plural, used in this expression

Some masculine Russian nouns referring to males end in -а or -я. They decline like many feminine-looking nouns, but they are still grammatically masculine.

Examples:

• дедушка = grandfather
• папа = dad
• дядя = uncle
• мужчина = man

So мой дедушка is correct, not моя дедушка, because the noun is masculine in meaning and agreement.

Because внука is the accusative case, and внук is an animate masculine noun.

In Russian, masculine animate nouns have:

• nominative: внук
• accusative: внука

Here, the grandson is the direct object of возит, so Russian uses the accusative:

• дедушка возит внука = grandfather takes/drives his grandson

This is a very important pattern:

• Я вижу брата
• Он знает учителя
• Мы любим кота

For animate masculine nouns, accusative usually looks like genitive.

Because this sentence describes movement toward a place.

With в:

• в + accusative = into / to
• в + prepositional = in / inside

So:

• в парк = to the park (direction)
• в парке = in the park (location)

Compare:

• Дедушка идёт в парк. = He is going to the park.
• Дедушка гуляет в парке. = He is walking in the park.

возить means to transport / take someone or something regularly, repeatedly, or back and forth, usually by some means of transport.

So возит here suggests a habitual action:

• По выходным мой дедушка возит внука в парк.
• On weekends, my grandfather takes/drives his grandson to the park.

This is different from a one-time trip.

This is a classic Russian verb pair.

• возить = to transport repeatedly / habitually / in various directions
• везти = to be transporting right now / in one конкретe trip / in one direction

So:

• Дедушка возит внука в парк по выходным.
  = He takes him there regularly.

• Сейчас дедушка везёт внука в парк.
  = Right now he is taking him to the park.

In your sentence, the idea is habitual, so возит is the natural choice.

Usually it suggests transporting someone by vehicle or at least not on foot.

For example, it can imply going:

• by car
• by bus
• by train
• by some other transport

If you specifically wanted takes on foot / leads, Russian would more likely use водит in some contexts:

• водить ребёнка в парк = to take a child to the park regularly

So возит often gives the feeling that the grandfather is driving or otherwise transporting the grandson.

Because Russian does not have articles like a or the.

So:

• дедушка can mean a grandfather, the grandfather, or grandfather, depending on context
• внука can mean a grandson, the grandson, or his grandson, depending on context

In this sentence, English naturally uses my grandfather and his grandson / the grandson, but Russian expresses that without articles.

The possessive мой helps make the subject specific:

• мой дедушка = my grandfather

Not always.

Russian often omits possessives when the relationship is obvious from context. So in some situations, you could hear:

• По выходным дедушка возит внука в парк.

That could still mean Grandpa takes his grandson to the park on weekends.

But мой дедушка makes it explicit that the speaker is talking about my grandfather, so it sounds natural and clear.

Yes. Russian word order is flexible because case endings show the grammatical roles.

The neutral order here is:

• По выходным мой дедушка возит внука в парк.

But you could also say things like:

• Мой дедушка по выходным возит внука в парк.
• Внука мой дедушка по выходным возит в парк.
• В парк мой дедушка возит внука по выходным.

These all keep roughly the same basic meaning, but the emphasis changes.

For learners, the original sentence is a very natural neutral version.

A repeated habit.

Two things show that:

1. по выходным = on weekends / every weekend
2. возит is the imperfective habitual verb

So the meaning is not this weekend my grandfather is taking his grandson to the park, but rather:

• My grandfather takes his grandson to the park on weekends.

It is in the nominative case, because it is the subject of the sentence.

The subject is the person doing the action:

• мой дедушка = the one who takes/drives

So the cases in the sentence are:

• по выходным = dative plural
• мой дедушка = nominative
• внука = accusative
• в парк = accusative after a motion preposition

This is a good sentence for seeing several common Russian case patterns at once.