Edit in JSFiddle

<div class="parent fixed">
    <div class="child">My parent has <code>height: 1px</code>!</div>    
</div>


<div class="parent">
    <div class="child">My parent has only <code>min-height: 10em</code>!</div>    
</div>

* {
    -webkit-box-sizing: border-box;
    -moz-box-sizing: border-box;
    box-sizing: border-box;
}

body {
    padding: 1em;
}

.parent {
    width: 49%;
    float: right;
    min-height: 10em;
    background: deepskyblue;
    font-family: Helvetica, Arial, sans-serif;
}

.parent.fixed {
    float: left;
}

.fixed {
    height: 1px;
}

.child {
    padding: .25em;
    background: tomato;
    height: 100%;
}