ane ha, gyouza wo yakinagara, tenpura no zyunbi mo site imasita.

は marks 姉 as the topic of the sentence.

Here, 姉は means something like as for my older sister... or my older sister was....
It sets up who the sentence is about, and then the rest of the sentence explains what she was doing.

If you used が instead, it would sound more like you are specifically identifying the one who was doing these actions. With は, the focus is more on describing her actions.

Yes, in a normal context, 姉 usually means my older sister when you are talking about your own family to someone else.

Japanese family words often work differently from English:

• 姉 = my older sister / older sister
• お姉さん = someone else’s older sister, or a more polite/friendly form

So even though English often needs my, Japanese often leaves that implied by context.

を marks the direct object of the verb 焼く.

So:

• 餃子を焼く = to cook/grill/pan-fry dumplings

Here, 餃子 is the thing receiving the action of 焼く, so it takes を.

ながら means while doing or as someone does.

So:

• 焼きながら = while cooking/grilling/pan-frying

It connects two actions done by the same person at the same time. In this sentence, the older sister was:

• cooking the dumplings
• while also preparing the tempura

A very important point: with ながら, the subject is normally the same for both actions.
So the person cooking the dumplings is also the person preparing the tempura.

It comes from the verb stem of 焼く.

Steps:

• dictionary form: 焼く
• verb stem: 焼き
• add ながら
• 焼きながら

This is the standard pattern:

• 書く → 書きながら = while writing
• 食べる → 食べながら = while eating
• 勉強する → 勉強しながら = while studying

So verb stem + ながら is the basic grammar pattern.

天ぷらの準備 means the preparation for tempura or getting the tempura ready.

The particle の connects the two nouns:

• 天ぷら = tempura
• 準備 = preparation

So 天ぷらの準備 is a noun phrase meaning tempura prep.

Depending on context, that could include things like:

• cutting ingredients
• mixing batter
• getting oil ready
• arranging things before frying

It does not necessarily mean the tempura was already being cooked.

も means also or too.

So 天ぷらの準備もしていました means:

• she was doing the tempura preparation too
• she was also preparing the tempura

This shows that preparing the tempura is an additional action besides cooking the dumplings.

A useful grammar point: も can replace particles like を in many cases.

So instead of saying:

• 天ぷらの準備をしていました

the sentence says:

• 天ぷらの準備もしていました

This adds the meaning of also.

していました is the past progressive / past continuous form. It means was doing.

So:

• しました = did
• していました = was doing

In this sentence, that matters because the sentence describes ongoing actions in progress in the past:

• she was cooking dumplings
• she was also preparing tempura

Using していました makes the scene feel like something that was happening over a span of time, not just a single completed action.

Because 準備 can function as a verbal noun with する.

So:

• 準備する = to prepare / to make preparations

And:

• 天ぷらの準備をする
• 天ぷらの準備もする

are natural ways to say to prepare for tempura.

In this sentence, 準備 is the noun and していました is the する part in polite past progressive form.

Usually it means the actions overlap, but not necessarily in a perfectly second-by-second way.

Here the idea is that she was multitasking:

• cooking dumplings
• and at the same time also working on tempura prep

So the actions are understood as happening in parallel, or at least in the same period of activity.

That is a very common question.

In Japanese, 焼く is often used for foods that are:

• grilled
• roasted
• baked
• pan-cooked on heat

For 餃子, especially the common pan-fried kind, Japanese normally says:

• 餃子を焼く

Even though in English you might say fry or pan-fry, Japanese still often uses 焼く.

So this is a normal and natural collocation:

• 餃子を焼く = to cook/pan-fry gyoza

Yes. The ending していました is in the polite ます style.

Compare:

• plain style: していた
• polite style: していました

So the sentence is describing the action politely, which is common in conversation, explanations, and writing aimed at being respectful or neutral.