Q:

Does changing resistance of resistor connected parallel to voltmeter change the output voltage of it?

Here is the circuit. 

My own logic says that the voltage output should stay the same even if the resistance of the resistor is increased because in parallel the voltage is equal.

A:

A voltmeter has a very high internal resistance, typically around $1M\Omega$. So when it is used in circuits with much smaller resistances $R$ of tens of $k\Omega$ or less, it draws very little current. Then practically all of the current from the battery flows through the battery and the load resistor $R$.
The internal resistance $r$ of a battery is typically $0.1-2\Omega$. In your circuit the emf $V$ of the battery is divided proportionately between $r$ and $R$ so the voltmeter will read $\frac{R}{R+r}V$. 
If $R\gg r$ then all of the emf of the battery appears across the load resistor, so the voltmeter will read $V$. Reducing load resistance $R$ reduces the reading of the voltmeter. When $R=r$ the voltmeter reads $\frac12V$. If $R \ll r$ then the reading on the voltmeter is very small, close to zero. 

