*From Computer* tab
----
we need to retrieve the `$attachment_parent_id`:

    add_filter('attachment_fields_to_edit', 'wpse_53600_remove_media_upload_fields', 10000, 2);
    function wpse_53600_remove_media_upload_fields( $form_fields, $post ) {
    	$our_page_id = 35;
        $post_id = !empty( $_GET['post_id'] ) ? (int) $_GET['post_id'] : 0;
    	$attachment_parent_id = !empty($post->post_parent) ? $post->post_parent : 0;
    	
        if( $post_id == $our_page_id || $attachment_parent_id == $our_page_id ){
            // remove unnecessary fields
            unset( $form_fields['image-size'] );
            unset( $form_fields['post_excerpt'] );
            unset( $form_fields['post_content'] );
            unset( $form_fields['url'] );
            unset( $form_fields['image_url'] );
            unset( $form_fields['align'] );
        }
        return $form_fields;
    }

----

*From URL* tab
----
looks like another approach is necessary:

    add_filter('type_url_form_media', 'wpse_53600_remove_from_url_fields', 10, 1);
    function wpse_53600_remove_from_url_fields($html) {
    	$our_page_id = 35;
    	$post_id = !empty( $_GET['post_id'] ) ? (int) $_GET['post_id'] : 0;
    
    	if( $post_id != $our_page_id)
    		return $html;
    	
    	// Used to hide the unwanted table rows
    	$display_none = 'style="display:none"';
    
    	$html = <<<HTML
    	 <p class="media-types">
    		<label><input type="radio" name="media_type" value="image" id="image-only" checked='checked' /> Image</label> &nbsp; &nbsp; <label><input type="radio" name="media_type" value="generic" id="not-image" /> Audio, Video, or Other File</label>
    	</p> 
    	<table class="describe ">
    		<tbody> 
    			<tr> <th valign="top" scope="row" class="label" style="width:130px;"> <span class="alignleft"><label for="src">URL</label></span> <span class="alignright"><abbr id="status_img" title="required" class="required">*</abbr></span> </th> <td class="field"><input id="src" name="src" value="" type="text" aria-required="true" onblur="addExtImage.getImageData()" /></td> </tr>
    			<tr> <th valign="top" scope="row" class="label"> <span class="alignleft"><label for="title">Title</label></span> <span class="alignright"><abbr title="required" class="required">*</abbr></span> </th> <td class="field"><input id="title" name="title" value="" type="text" aria-required="true" /></td> </tr>
    			<tr class="not-image" {$display_none}><td></td><td><p class="help">Link text, e.g. &#8220;Ransom Demands (PDF)&#8221;</p></td></tr>
    			<tr class="image-only" {$display_none}> <th valign="top" scope="row" class="label"> <span class="alignleft"><label for="alt">Alternate Text</label></span> </th> <td class="field"><input id="alt" name="alt" value="" type="text" aria-required="true" /> <p class="help">Alt text for the image, e.g. &#8220;The Mona Lisa&#8221;</p></td> </tr>
    			<tr class="image-only" {$display_none}> <th valign="top" scope="row" class="label"> <span class="alignleft"><label for="caption">Image Caption</label></span> </th> <td class="field"><input id="caption" name="caption" value="" type="text" /></td> </tr>
    			<tr class="align image-only" {$display_none}> <th valign="top" scope="row" class="label"><p><label for="align">Alignment</label></p></th> <td class="field"> <input name="align" id="align-none" value="none" onclick="addExtImage.align='align'+this.value" type="radio" checked="checked" /> <label for="align-none" class="align image-align-none-label">None</label> <input name="align" id="align-left" value="left" onclick="addExtImage.align='align'+this.value" type="radio" /> <label for="align-left" class="align image-align-left-label">Left</label> <input name="align" id="align-center" value="center" onclick="addExtImage.align='align'+this.value" type="radio" /> <label for="align-center" class="align image-align-center-label">Center</label> <input name="align" id="align-right" value="right" onclick="addExtImage.align='align'+this.value" type="radio" /> <label for="align-right" class="align image-align-right-label">Right</label> </td> </tr>
    			<tr class="image-only" {$display_none}> <th valign="top" scope="row" class="label"> <span class="alignleft"><label for="url">Link Image To:</label></span> </th> <td class="field"><input id="url" name="url" value="" type="text" /><br /> <button type="button" class="button" value="" onclick="document.forms[0].url.value=null">None</button> <button type="button" class="button" value="" onclick="document.forms[0].url.value=document.forms[0].src.value">Link to image</button> <p class="help">Enter a link URL or click above for presets.</p></td> </tr>
    			<tr class="image-only"> <td></td> <td> <input type="button" class="button" id="go_button" style="color:#bbb;" onclick="addExtImage.insert()" value="Insert into Post" /> </td> </tr>
    			<tr class="not-image"> <td></td> <td> <input type="submit" name="insertonlybutton" id="insertonlybutton" class="button" value="Insert into Post" /> </td> </tr>
    		</tbody>
    	</table>
    HTML;
    
    	return $html;
    }