Můj pes miluje hrát si v parku.

It is understandable, but it does not sound like the most natural everyday Czech.

A native speaker would more likely say:

Můj pes si rád hraje v parku.

Why?

• Czech often uses rád + finite verb for the idea of likes/loves doing something
• si usually does not stay after the infinitive in a full sentence the way dictionary forms do
• the original sentence sounds a bit like a word-for-word translation from English

So as a learner, it is useful to understand the sentence, but it is even more useful to know the more idiomatic version.

Můj means my, and it has to agree with the noun it describes.

Here, pes is:

• masculine animate
• singular
• nominative because it is the subject of the sentence

So the correct form is můj pes.

Compare:

• můj pes = my dog
• moje kočka = my cat
• moje auto = my car

Miluje is the 3rd person singular present tense of milovat = to love.

It matches můj pes because my dog is:

• 3rd person
• singular

So:

• já miluji / miluju = I love
• ty miluješ = you love
• on/ona/ono miluje = he/she/it loves

Here, pes takes the same verb form as he/it, so you get miluje.

Because Czech infinitives do not use a separate word like English to.

In English:

• to play

In Czech:

• hrát

So hrát already means to play.
The to is built into the infinitive form itself.

This is true for Czech verbs in general:

• být = to be
• mít = to have
• jít = to go
• hrát = to play

This is a very important question.

hrát often means:

• to play a game or sport
• to play an instrument
• to play a role

Examples:

• hrát fotbal = to play football
• hrát na kytaru = to play the guitar
• hrát Hamleta = to play Hamlet

hrát si means:

• to play around
• to have fun playing
• the kind of play children or animals do

So for a dog in a park, hrát si is the right idea.

A dog usually:

• si hraje not
• hraje

In hrát si, si is part of the verb expression. It does not have a neat one-word English translation.

You should usually learn hrát si as a whole unit meaning:

• to play in the sense of enjoying playful activity

A useful point for learners:

• in the dictionary, you often see hrát si
• in real sentences, si often moves earlier because it is a clitic

That is why Czech more naturally says:

• Pes si hraje rather than
• Pes hraje si

This is one reason the original sentence feels awkward.

Because v means in or at, and here it shows location.

So:

• v parku = in the park

After v in this meaning, Czech uses the locative case.

That is why:

• park becomes parku

So:

• park = park
• v parku = in the park

Because do parku means movement into the park, while v parku means being in the park.

Compare:

• Jdu do parku. = I am going to the park.
• Jsem v parku. = I am in the park.

In your sentence, the dog is playing in the park, not moving to it, so v parku is the correct choice.

Because Czech has no articles.

English distinguishes:

• a park
• the park

Czech usually just says:

• park
• v parku

Context tells you whether the meaning is:

• a park
• the park
• parks in general

So v parku can mean in a park or in the park, depending on the situation.

Yes. Czech word order is more flexible than English word order.

However, not every order sounds equally natural, and small words like si have special placement rules.

A neutral, natural version is:

• Můj pes si rád hraje v parku.

You can move things for emphasis, for example:

• V parku si můj pes rád hraje. = In the park, my dog likes to play.
• Můj pes si v parku rád hraje.

So word order can change, but you still have to respect natural Czech patterns, especially with si.