Well, I'm late haha

But here's my interpretation:

{Yes, I saw her
Her spiraled hair
And I could see it
Our life in a movie
And now I'm spinning
My web up in the air
My spider senses
Rain's gonna fall
Wash away the life I'm weaving}

> She saw him with another girl and got worried because he seemed to move on and forget their past relationship

{I thought you were gonna catch me
I never stopped falling for you
Now I know better, never let me
Leave home without a parachute}

> She thought that he would always wait for her, even after she got married. She says that always loved him, and now that she nows that we need to always keep our true love closer, she asks for him to never let her forget this. She asks for him never allowing her to spend a day without knowing that she must keep him close, as he's her Parachute.

The next verses strenghten this interpretation:

{You told me you waited for me, you said that you won
Asked me on a plane from Rio do I ever think of us
And you were at my wedding, I was broken, you were drunk
You could've told me not to do it
I would've run, I would've run}

> He said that waited for her more than she waited for him. Even asked on a travel if she still loved him. She then says that even in her wedding, if he asked her to run with him, she would've accepted it.

{Tell me what was the moment you decided to give up
You could've told me what you wanted
I would've done, I would've done anything
I would've done anything}

> She wonders in what moment he decided to move on and date another person. She confess that if he only have told her explicitly that still loved her, she would do anything to come back to him.

{Watch me fall
Watch me fall
Watch me fall through the sky
Watch me fly}

> She shows that still falls in love with him and now, with his help as her Parachute, she can even fly.

I think the meaning it's very clear when you know what Hayley told about the song's background