Play and wait for Animation/Animator to finish playing

While both answers should work, another method of doing this with coroutine and the IsPlaying function. You use the coroutine solution if you also want to perform other task after the animation.

For the Animation system:

The old Unity animation playback system. This should not be used in your new Project unless you are still using old Unity version.

IEnumerator playAndWaitForAnim(GameObject target, string clipName)
{
    Animation anim = target.GetComponent<Animation>();
    anim.Play(clipName);

    //Wait until Animation is done Playing
    while (anim.IsPlaying(clipName))
    {
        yield return null;
    }

    //Done playing. Do something below!
    Debug.Log("Done Playing");
}

For the Animator system

This is the new Unity animation playback system. This should be used in your new Project instead of the Animation API. In terms of performance, it’s better to use the Animator.StringToHash and compare the current state by hash number instead of the IsName function which compares string since the hash is faster.

Let’s say that you have state names called Jump, Move and Look. You get their hashes as below then use the function for playing and waiting for them them below:

const string animBaseLayer = "Base Layer";
int jumpAnimHash = Animator.StringToHash(animBaseLayer + ".Jump");
int moveAnimHash = Animator.StringToHash(animBaseLayer + ".Move");
int lookAnimHash = Animator.StringToHash(animBaseLayer + ".Look");

public IEnumerator PlayAndWaitForAnim(Animator targetAnim, string stateName)
{
    //Get hash of animation
    int animHash = 0;
    if (stateName == "Jump")
        animHash = jumpAnimHash;
    else if (stateName == "Move")
        animHash = moveAnimHash;
    else if (stateName == "Look")
        animHash = lookAnimHash;

    //targetAnim.Play(stateName);
    targetAnim.CrossFadeInFixedTime(stateName, 0.6f);

    //Wait until we enter the current state
    while (targetAnim.GetCurrentAnimatorStateInfo(0).fullPathHash != animHash)
    {
        yield return null;
    }

    float counter = 0;
    float waitTime = targetAnim.GetCurrentAnimatorStateInfo(0).length;

    //Now, Wait until the current state is done playing
    while (counter < (waitTime))
    {
        counter += Time.deltaTime;
        yield return null;
    }

    //Done playing. Do something below!
    Debug.Log("Done Playing");

}

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For a solution specifically for your particular problem with the collision callback function (OnTriggerEnter), there are two possible ways to do that:

1.Start a coroutine function to play the animation after trigger detection:

void OnTriggerEnter(Collider other)
{
    if (other.gameObject.name == "OnTop Detector")
    {
        Debug.Log("On Top of Platform");
        GameObject findGo = GameObject.Find("ThirdPersonController");
        GameObject findGo1 = GameObject.Find("Elevator");
        findGo.transform.parent = findGo1.transform;
        target = GameObject.Find("Elevator");
        StartCoroutine(playAnim(target));
    }
}

IEnumerator playAnim(GameObject target)
{
    Animation anim = target.GetComponent<Animation>();
    anim.Play("Up");

    //Wait until Up is done Playing the play down
    while (anim.IsPlaying("Up"))
    {
        yield return null;
    }

    //Now Play Down
    anim.Play("Down");
}

OR

2.Make the OnTriggerEnter function a coroutine(IEnumerator) instead of void function:

IEnumerator OnTriggerEnter(Collider other)
{
    if (other.gameObject.name == "OnTop Detector")
    {
        Debug.Log("On Top of Platform");
        GameObject findGo = GameObject.Find("ThirdPersonController");
        GameObject findGo1 = GameObject.Find("Elevator");
        findGo.transform.parent = findGo1.transform;
        target = GameObject.Find("Elevator");
        Animation anim = target.GetComponent<Animation>();
        anim.Play("Up");

        //Wait until Up is done Playing the play down
        while (anim.IsPlaying("Up"))
        {
            yield return null;
        }

        //Now Play Down
        anim.Play("Down");
    }
}